1. Introduction

State estimator is an algorithm that process raw and redundant conventional meter readings and other information i.e. network topology, circuit breaker status etc. to estimates the state of a power system [1, 2, 3]. State estimation is one of the basic tools used to ensure that the system is operational in secure mode and all constraints are satisfied. The installed systems at control center of electrical utilities process different measurements from different types of sensors and meters to estimate the overall operating condition of a power system. The measurements which are recorded wrongly because of the large noise, aging of a meter or calibration issue etc. are referred as bad-data [4] that affects the estimation process and results in the wrong estimation of the system's state variables (voltage magnitude and angle). Bad-data is mainly classified into two categories: (a) single bad-data and (b) multiple bad-data [5]. Multiple bad-data mostly occurs in very large systems and is strongly correlated to each other that poses a huge impact on the results of state estimation.

Weighted Least Square (WLS) is most popular state estimator deployed in electrical utilities and typically intakes reading from conventional meters including power flow meter, power injection meter, and voltage magnitude meter. The mathematical formulation of WLS is simple and has less computational burden, however, it is a non-robust state estimator because WLS fails to produce reliable estimation results even in the presence of single bad-data.

Weighted Least Absolute Value (WLAV) is a robust estimator, compared with WLS, however, it is susceptible to leverage points for certain configurations of meter distribution in a systems [6]. WLAV poses computational burden for large power systems that limits it utilization. For numerical stability and computational efficiency in linear programming (LP), the scaling technique is widely employed and proved an efficient tool [7] provided that scaling helps in reducing the effect of leverage points. In [8], a robust WLAV-T estimator is proposed to mitigate the effect of leverage points based on optimal transformation of associated rotation angles and scaling factor in systematic way, compared to heuristic approach. Further, the WLAV possesses auxiliary variables that reduce the convergence rate of the estimator, so Weighted Linear Least Square (WLLS) is proposed in [9] that possessed less number of variables than WLAV estimator and linear objective function.

Traditional state estimation is going through essential developments due to the innovation of Phasor Measurement Units (PMUs). State estimation problem is more easily formulated when there are adequate number of only PMUs installed in a power system because there is linear relationship between PMU's phasor measurements and state variables [10]. Least Absolute Value (LAV) estimator is computationally efficient and competitive with WLS if only phasor measurements are provided for state estimation and the strategic scaling method may be used to avoid biasness of leverage points [11, 12]. In [10], the authors investigated the performance comparison between WLS and LAV when only PMUs were used in the presence of different bad-data scenarios and applied scaling technique to maintain the robustness of LAV. The accuracy, synchronization and redundancy of state estimation is improved by incorporating PMU phasor measurements with conventional SCADA measurement into the existing state estimator, however, there are many challenges in this implementation that are addressed with proposed solution [13]. In [14], the authors proposed a method to incorporate conventional SCADA having slow refresh rate of measurement and PMU phasor measurements with fast refresh rate in WLS estimator to keep track of the varying state of the power network. A hybrid state estimator [15] integrated with conventional Supervisory Control and Data Acquisition (SCADA) and PMU's phasor measurements is utilized for observability analysis and state estimation of a power system. In this hybrid estimator, there is switching between WLS and LAV estimator depending upon the availability of measurements from conventional SCADA or PMU's phasor measurements. The availability of measurements further depends upon the variation in refresh rate of SCADA and PMU's phasor measurements. The basic idea of hybrid state estimator is presented in [16]. A two stage linear estimator with only PMUs is proposed in [17] that is not only robust estimator but also computationally efficient because of even distribution of processing burden among different areas.

A PMU measures voltage phasor at a bus and current phasor of all incoming and outgoing flows at substation where it has been installed [18]. A modified non-linear WLS estimator incorporating voltage and current phasors in rectangular and polar coordinates is presented in [19] and both approaches have been compared and evaluated on IEEE-14 bus system with different bad-data scenarios. In [20], a two-stage state estimator combining both SCADA and PMU is proposed and authors have claimed higher estimation accuracy over only conventional SCADA meters. An approach to integrate the PMU technology into the existing SCADA systems to improve the accuracy of state variable is proposed, however, the proposed approach has been tested on very small power system without providing solid mathematical formulation [21]. Another technique to combine both SCADA and \(\mu\)PMU meters in a distribution system state estimation is presented, however, the authors have proposed this approach for WLS, which is a non-robust state estimator [22]. In [23], the authors have proposed an approach to integrate both SCADA and \(\mu\)PMU meters together, however, the proposed approach is applied for network topology analysis and tested on IEEE-33 bus system only. Furthermore, the authors have not provided computational efficiency of the proposed approach.

Least Measurement Rejected (LMR) is a robust estimator which associate a tolerance range with each provided measurement. LMR is solved using mixed integer programming approach and it rejects unreliable, corrupted or wrongly recorded measurements during estimation process and it is not susceptible to leverage points [24]. LMR is simple and effective state estimation approach. In [25], the authors have proposed an iterative tuning approach to choose the appropriate tolerance value of LMR for a certain measurement configuration. The authors of [26] have proposed a novel approach to tune and select the best value for tolerance parameter of LMR. In literature, the different authors have integrated PMUs into existing SCADA system in different estimation algorithms e.g. WLS, WLAV, however, PMU's phasor measurements have not been incorporated in LMR which is also the novelty of our work.

Though, the above authors were successful in implementing the integration of SCADA and PMUs, however, many of them have applied their proposed approaches on non-robust estimator or the chosen test case was small power system. Furthermore, many authors could not provide information about computational efficiency and number to iterations required to complete the estimation process. Our proposed technique has been applied on two robust state estimators: WLAV and LMR and evaluated on IEEE-30 and IEEE-118 bus systems in terms of state variables (voltage magnitude and angle), computational efficiency and number of iterations. Further, the objective of this paper is to improve the state estimation accuracy in the presence of different bad-data scenarios (single and multiple) by incorporating the voltage and current phasor from PMUs. The tolerance parameter of LMR estimator is chosen properly to reject large errors in conventional measurements. The final approach of achieving the objective is the proper selection of the PMUs locations to ensure best performance and accuracy.

The paper work is organized as follows. The section 2 covers the formulation and details about inclusion of voltage and current phasors in the state estimators and modified mathematical model of the state estimators. Performance validation of proposed technique is presented in section 3. Finally, section 4 concludes the paper with a brief summary.

2. Inclusion of Voltage and Current Phasors into Robust State Estimators

Once a PMU is installed at a bus in a power system, the voltage phasor of the bus and current phasor of all the branches connected to the bus is measured accurately. In this paper, these PMU's phasors are incorporated into the existing robust estimators and it was expected to achieve higher state estimation accuracy, compared with conventional measurements. When a phasor measurement is included in a state estimator, the weight of the phasor measurement must be increased because the PMU's measurement is highly accurate [10]. In WLAV estimator, there is a weight matrix where any measurement is assigned a specific weight corresponds to its accuracy, however, there is no weight matrix or covariance matrix in LMR estimator. The simulation analysis of our paper reveals that reducing the tolerance parameter of LMR associated with a measurement is equivalent to increasing the weight of the measurement in WLAV.

It is required to build a relationship between branch current flows in transmission lines and state variables to incorporate the current phasor in the state estimators. The proposed model will include all transmission lines and transformers between buses. An ampere flow in a branch is expressed as given below [27]:

where \(I_{ij,real}\) and \(I_{ij, imag}\) are rectangular components of the branch current flowing between bus \(i\) and \(j\), \(V_{i}\), \(V_{j}\), \(\theta_{i}\) and \(\theta_{j}\) are voltage magnitude and phase angle of bus \(i\) and \(j\) respectively, \(g_{ij}\) and \(b_{ij}\) are conductance and susceptance between bus \(i\) and \(j\) respectively, \(g_{si}\) and \(b_{si}\) are shunt conductance and susceptance respectively.

Practically, a PMU provides ampere measurements in polar coordinates rather than rectangular coordinates and can referred as direct measurements. However, it is better to use current phasors in rectangular coordinates, because the power flow measurements and power injection measurements are already in rectangular coordinates. In this regard, the ampere measurements should be converted to rectangular coordinates and utilized for state estimation. If the direct measurements are converted to rectangular coordinates, then error covariance values must be translated for rectangular coordinates. The relation between direct and indirect measurement is given by following equations:

where \(I_{ij}\) is branch current magnitude between bus \(i\) and \(j\), \(\theta_{I_{ij}}\) is branch current phase angle between bus \(i\) and \(j\). The standard deviation of translated measurements can be calculated from below given Equations 5 and 6 [19], [28]. \begin{equation*} \sigma_{I_{ij,real}}= \sqrt{\big(cos(\theta_{I_{ij}})\big)^2 \sigma^2_{I_{ij}}+\big(I_{ij}sin(\theta_{I_{ij}})\big)^2 \sigma^2_{\theta_{I_{ij}}}} \end{equation*} \begin{equation*} \sigma_{I_{ij,imag}}= \sqrt{\big(cos(\theta_{I_{ij}})\big)^2 \sigma^2_{I_{ij}}+\big(I_{ij}sin(\theta_{I_{ij}})\big)^2 \sigma^2_{\theta_{I_{ij}}}} \end{equation*} where \(\sigma_{I_{ij,real}}\) and \(\sigma_{I_{ij,imag}}\) are standard deviation of real and imaginary parts of current flows respectively, \(\sigma_{I_{ij}} = 0.02 \ p.u \) and \(\sigma_{\theta_{I_{ij}}} = 0.0017 \ rad \) and are standard deviation of \(I_{ij}\) and \(\theta_{I_{ij}}\) respectively.

2.1. Modified Weighted Least Absolute Value (WLAV) with Voltage and Current Phasors

In this section, the mathematical formulation of modified WLAV incorporated with voltage and current phasors is presented. Usually, LP solving approach like simplex method or interior point method is used for WLAV. The performance of WLAV is very good for rejection of bad-data, however, it fails to provide reliable estimation results in the presence of leverage points [4]. There are numerous methods to identify leverage measurements [29] which is not the scope of this work.

A power system consists of \(n\) buses with a specific network topology where \(m\) meters are placed at different bus and branch locations to provide measurements from meters to state estimators through Remote Terminal Unit (RTU). The measurement vector \(z\) of size \((m\) x \(1)\) is fed into a state estimator to obtain the state variable vector \(x\) of size \(n=(2n\) x \(1)\). The non-linear function relating measurements to system state variables is given below:

where \(h(x)\) is the non-linear function relating measurements with the state vectors, and \(e\) is the measurement error vector. Using the first order approximation for the Equation \ref{7} , it can be written around some operating point \(\hat{x}\) as where \(P_{ij}\) and \(Q_{ij}\) are real and reactive power flow between bus \(i\) and \(j\), \(P_{inj}\) and \(Q_{inj}\) are real and reactive power injection between bus \(i\) and \(j\) respectively, \(Vm_{i}\) and \(\theta_{i}\) are voltage magnitude and phase angle between at bus \(i\), and \(H\) is the modified Jacobian matrix as voltage and current phasors from PMUs are incorporated in it. The minimization objective function for WLAV is where \(\sigma_{i}\) is standard deviation of \(ith\) measurement. The standard deviation of PMU's measurements will be given different from the measurements of SCADA meters because PMU's measurements are considered more accurate, compared with measurements from SCADA. The values of standard deviation of SCADA and PMU's measurements are given in section 3. The minimization problem in 14 can be transformed into LP problem Subject to \begin{equation*} A\cdot Y = b \end{equation*} \begin{equation*} Y\geq0 \end{equation*} where \begin{equation*} c^T = [0_{n} \ 0_{n} \ \frac{1}{\sigma_{p}} \ \frac{1}{\sigma_{p}}] \end{equation*} \begin{equation*} 0_{n}=[0,\ldots,0] \end{equation*} \begin{equation*} \sigma_{p}=[\sigma_{1},\sigma_{2},\ldots,\sigma_{m}] \end{equation*} \begin{equation*} Y^T = [x_{u} \ \ x_{v} \ \ u \ \ v] \end{equation*} \begin{equation*} A = [H \ -H \ I_{m} \ -I_{m}] \end{equation*} \begin{equation*} b = \triangle z \end{equation*} where \(x_{u}\) and \(x_{v}\) are the components of state variables and both of these variables are size of \(n=(2N-1)\), \(N\) is number of buses, \(I_{m}\) is identity matrix of size \(m \ x \ m\), and \(m\) is number of measurements. The difference between \(x_{u}\) and \(x_{v}\) will provide values of state variables at each iteration.

2.2. Modified Least Measurement Rejected with Voltage and Current Phasors

A robust state estimation technique was proposed using mixed integer programming in [24], which correlates a specific tolerance value to each measurement in a power system. For each measurement, a tolerance value is defined which helps the estimator to reject unreliable or corrupted measurements during the estimation process. This robust estimation approach is invulnerable to leverage points even in pathological cases. The upper and lower tolerance values can be asymmetrical or symmetrical e.g. a power injection meter of 10MW may have upper tolerance value of 0.5MW and lower tolerance value of 0.75MW, the measurement value taken from the power injection meter between (10.5MW and 9.25MW) will be considered as good and reliable. However, if the measurement value lies outside the specified range, then it will be suspected as corrupted or bad-data. This is an effective and trivial estimation approach and known as Least Measurement Rejected (LMR) and it is a variant of Least Median of Square (LMS). LMR minimizes the number of rejected measurements within a defined tolerance and its objective function is given below: where \(h_{i}(x)\) is measurement equation of \(ith\) meter, \(t_{i}\) is tolerance value of \(ith\) measurement, \(k_{i}\) is a binary number indicating whether the measurement error is within a specified tolerance limit or not, and \(M\) is arbitrary large scalar value. The Equations 16 and 17 have been transformed into mixed integer programming problem and the formulation can be written as: Subject to \begin{equation*} A.Y = b \end{equation*} where \begin{equation*} c^T =[0_{n} \ \ 1_{m}] \end{equation*} \begin{equation*} A= \begin{bmatrix} h & -M \\ h & -M \end{bmatrix} \end{equation*} \begin{equation*} B^T = [b+t \ b-t] \end{equation*} \begin{equation*} b=\triangle z \end{equation*} \begin{equation*} Y^T = [\triangle x \ \ k] \end{equation*} where, \(n\) is number of state variables, \(H\) is modified Jacobean matrix explained in Equation 9, \(M\) is a vector of arbitrary large scaler value, \(t\) is a vector that contains tolerance values of all provided measurements, \(b\) is a vector of difference between estimated and loaded measurement values and \(\triangle x\) is a vector of change is state variables.

3. Performance Validation

The method of voltage and current phasors inclusion into the existing robust state estimators is discussed in section 2 and is evaluated in the presence of different bad-data scenarios. Five different types of bad-data scenarios are considered in this paper, such as: (a) single bad-data (SBD) as power flow meter, (b) single bad-data (SBD) as a power injection meter, (c) single bad-data (SBD) as a voltage magnitude meter, (d) multiple non-interacting bad-data (MNIBD), and (e) multiple interacting bad-data (MIBD) as a combination of power flow meter, power injection meter, and voltage magnitude meter. This paper also investigates the importance of locating PMUs in a power system, however, optimal placement of PMUs is not the scope of this paper. Only two PMUs are placed heuristically in the considered test cases. The locations for bad-data scenarios is carefully chosen to ensure that the meter location selected for bad-data is not a critical location. The performance of robust state estimators is compared in terms of absolute error between actual and estimated voltage magnitudes and actual and estimated voltage angles, and computational efficiency. The actual values of voltage magnitude and angle are taken from load flow solution. The units for absolute voltage magnitude and angle are per unit and degrees respectively. The lower the absolute error, the better is the performance of the estimator. The proposed approach has been applied on IEEE-30 and IEEE-118 bus systems. The test cases are prepared on observable heuristic approach and each test case possesses different set of measurement types and different redundancy level. MATPOWER package is deployed to complete the execution of state estimation algorithm in MATLAB. WLAV and LMR have been solved using \(lp\_solve\) package and mixed integer programming approach in MATLAB R2015a respectively. The simulation is performed on Intel Core i5, 4th generator processor with 4GB RAM. For LMR, \(M = 50000\) has been used for the IEEE-30 and IEEE-118 bus systems. The details of measurement types before and after the inclusion of PMUs are presented in Table 1 and Table 2 respectively.

Table 1. Measurement types.

Measurement type	Description
\(PF_{i-j}\)	Real power flow from bus \(i\) to \(j\)
\(PF_{j-i}\)	Real power flow from bus \(j\) to \(i\)
\(PG_{inj}\)	Real power injection at bus \(i\)
\(QF_{i-j}\)	Reactive power flow from bus \(i\) to \(j\)
\(QF_{j-i}\)	Reactive power flow from bus \(j\) to \(i\)
\(QG_{inj}\)	Reactive power injection at bus \(i\)
\(Vm_{i}\)	Voltage magnitude at bus \(i\)

Table 2. Measurement types from PMUs.

Measurement Type	Description
\(\theta_{i}\)	Voltage Angle
\(I_{i-j,real}\)	Real current flow from bus i to j
\(I_{i-j,imag}\)	Imaginary current flow from bus i to j
\(I_{j-i,real}\)	Real current flow from bus j to i
\(I_{j-i,imag}\)	Imaginary current flow from bus j to i

The values of standard deviation for SCADA and PMU's measurements are provided in Table 3 and the standard deviations of current phasors are calculated using Equations 5 and 6.

Table 3. Measurements standard deviation.

Measurement type	Standard deviation
	Without PMU	With PMU
\(PF_{i-j}, PF_{j-i}, PG_{inj}\)	0.02	0.0002
\(QF_{i-j}, QF_{j-i}, QG_{inj}\)	0.04	0.0004
\(Vm_{i}\)	0.01	0.0001

Usually, the values of standard deviation are used to calculate the weighting factor for WLAV. After the inclusion of PMU's measurements in WLAV, the weights of those specific meter readings have to be increased. The simulated measurements for PMU's phasors are considered as the actual load flow values obtained from the power flow solution. In LMR, since there is no weighting factor in its mathematical formulation, the PMU's measurements are taken as actual load flow values and the tolerance value of those specific measurements has to be set as zero. In this case, the LMR provides the excellent estimation results, compared with other state estimators.

3.1. IEEE 30 Bus System

The meter distribution details for the test case are listed in Table 4. The reference bus is slack bus '1' having zero-degree phase angle. The test case considered in this paper has only 126 SCADA meters. There are total 14 voltage magnitude meters in the test case, so there is wide-ranging choice for PMUs placement. In all cases for PMUs placement, one PMU is placed fixed at bus '1' to act as the reference PMU. The second location for PMU placement is chosen heuristically and it is ensured while placing PMU that the system is completely observable. All the critical measurements and sets are identified in the test case before simulating a bad-data at different types of meters.

Table 4. IEEE 30 Bus - Meter distribution.

Measurement type	Number of measurements
Real Power Flows	41
Real Power Injection	16
Reactive Power Flows	40
Reactive Power Injection	15
Voltage Magnitude	14

Computational time to complete a process or execute an algorithm is one of the major constraints in today's computing resources. The computation efficiency of WLAV and LMR for IEEE-30 bus system is shown in Table 5 and it is clearly reflected from the results that LMR requires lower computational time and a less number of iterations than WLAV. It can be noticed that the maximum time required by LMR to complete state estimation is lower than minimum time required by WLAV.

Table 5. IEEE 30 Bus - Computation efficiency of the state estimators.

State estimator	Convergence time (sec)		Number of iterations
	Min	Max
WLAV	0.31	0.36	4
LMR	0.17	0.28	2

The results for absolute voltage magnitude (AVM) error with different bad-data scenarios are shown in Table 6. In the first column, `NO PMU' means that there are only SCADA meters available in the test case. As already explained, one PMU is kept fixed at slack bus and another is relocated at different buses to get estimation results and highlight the importance of PMU placement. PMU-3 means that one PMU is fixed at slack bus and another is placed at bus 3.

It can be noticed from the results in Table \ref{table:6} that AVM error of 'NO PMU' is higher than all PMU placement cases i.e. PMU-3, PMU-5, PMU-10 and PMU-21 for both WLAV and LMR. However, any optimal location cannot be suggested as the optimal placement of PMUs is not the scope of this work. In this paper, the objective was to incorporate PMUs in existing state estimators and to compare the performance of WLAV and LMR in terms of computational efficiency and accuracy.

It can be observed from the results that inclusion of PMUs has improved the estimation accuracy of both WLAV and LMR estimators in all PMU placements which is reflected by the lower value of AVM error in all PMU placement cases than AVM of `No PMU' case and LMR has better state estimation accuracy than WLAV.

Table 6. IEEE 30 Bus - Absolute voltage magnitude (AVM) error.

PMUlocation	White Noise		SBD aspower flowat \(PF_{2-5}\)		SBD as powerinjectionat \(PG_{2}\)		SBD as voltage magnitude \(Vm_{12}\)		MNIBD at \(PF_{2-4}\), \(PG_{5}\), \(Vm_{12}\)		MNIBD at \(PF_{6-9}\) , \( PG_{24}\), \(Vm_{12}\)
	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR
NO PMU	0.0498	0.0200	0.0499	0.0200	0.0490	0.0212	0.0841	0.0664	0.0856	0.0559	0.0731	0.0584
PMU-3	0.0394	0.0100	0.0394	0.0098	0.0328	0.0093	0.0655	0.0187	0.0650	0.0160	0.0500	0.0126
PMU-5	0.0397	0.0104	0.0397	0.0104	0.0397	0.0198	0.0627	0.0201	0.0661	0.0200	0.0398	0.0139
PMU-10	0.0207	0.0108	0.0207	0.0112	0.0205	0.0121	0.0311	0.0215	0.0230	0.0212	0.0242	0.0207
PMU-21	0.0217	0.0169	0.0217	0.0169	0.0231	0.0163	0.0260	0.0219	0.0255	0.0247	0.0325	0.0197

The results for absolute voltage angle (AVA) error with different bad-data scenarios are shown in Table 7 and the angles are measured in degrees. It can be noticed from that AVA error of all PMU placement cases is lower than 'NO PMU' case for both WLAV and LMR. It is clearly reflected that inclusion of PMUs in the existing state estimators have not only improve voltage magnitude but also improved voltage angle for both WLAV and LMR. However, LMR has higher estimation accuracy as compared with WLAV in all PMU placement cases.

Table 7. IEEE 30 Bus - Absolute voltage angle (AVA) error.

PMUlocation	White Noise		SBD aspower flowat \(PF_{2-5}\)		SBD as powerinjectionat \(PG_{2}\)		SBD as voltage magnitude \(Vm_{12}\)		MNIBD at \(PF_{2-4}\), \(PG_{5}\), \(Vm_{12}\)		MNIBD at \(PF_{6-9}\) , \( PG_{24}\), \(Vm_{12}\)
	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR
NO PMU	2.8236	0.9927	2.9621	0.9927	2.2643	1.6630	3.5216	1.9322	4.0834	3.5875	4.4249	2.4865
PMU-3	1.1620	0.5947	1.1620	0.6011	1.2285	0.5570	1.1042	0.5036	1.1199	0.4136	1.1445	0.6171
PMU-5	1.0387	0.4735	1.0387	0.4715	1.2676	1.0270	1.1646	0.3330	1.0109	0.3414	1.0387	0.9417
PMU-10	0.6354	0.3040	0.6354	0.3061	0.5994	0.4121	1.0486	0.4871	0.7861	0.5908	0.5848	0.5200
PMU-21	0.8452	0.2509	0.8452	0.2313	1.0931	0.9487	0.8626	0.3419	0.9876	0.4605	0.8827	0.4030

3.2. IEEE 118 Bus System

The meter distribution details for the test case have been shown in Table 8. The test case considered in this paper has only 441 SCADA meters with a global redundancy of 1.87. There are total 61 voltage magnitude meters, so there is wide choice for PMU placement. The PMUs are installed on randomly chosen buses. In IEEE-118 bus system, the slack bus is 69 and voltage phase angle is 30 degrees. One PMU is kept fixed at bus reference bus 69 and second PMU is relocated heuristically at different locations. It also has been ensured that system is completely observable, and all the critical measurements and sets have been identified before simulating bad-data scenarios on different types of SCAD measurements.

Table 8. IEEE 118 Bus - Measurement distribution.

Measurement type	Number of measurements
Real Power Flows	134
Real Power Injections	55
Reactive Power Flows	134
Reactive Power Injections	56
Voltage Magnitude	61

The computational efficiency and number of iterations to complete state estimation are shown in Table 9. It can be seen from the presented results that LMR has lower computational time as compared with WLAV. However, both state estimators have same number of iterations but the overall computational efficiency of LMR is higher than WLAV as it requires lower completion time in seconds.

Table 9. IEEE 118 Bus - Computation efficiency of the state estimators.

State estimator	Convergence time (sec)		No. of iterations
	Min	Max
WLAV	6.40	9.17	6
LMR	3.73	4.26	6

The results of AVM error for IEEE 118 bus system are shown in Table 10. It can be noticed from these results that inclusion of PMUs has improved the state estimation accuracy of both WLAV and LMR. However, LMR provided higher state estimation accuracy as compared with WLAV in all PMU placement cases.

Table 10. IEEE 118 Bus - Absolute voltage magnitude (AVM) error.

PMUlocation	White Noise		SBD aspower flowat \(PF_{23-32}\)		SBD as powerinjectionat \(PG_{66}\)		SBD as voltage magnitude at \(Vm _{36}\)		MNIBD at \(PF_{4-11}\), \(PG_{16}\), \(Vm_{3}\)		MNIBD at \(PF_{34-36}\) , \( PG_{33}\), \(Vm_{18}\)
	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR
NO PMU	0.2792	0.1976	0.2602	0.1775	0.2700	0.1816	0.2791	0.1976	0.2841	0.2090	0.2883	0.1816
PMU-5	0.2035	0.1622	0.1910	0.1189	0.1992	0.1689	0.2034	0.1621	0.2135	0.1702	0.2135	0.1157
PMU-12	0.2199	0.1786	0.2111	0.1715	0.2154	0.1811	0.2198	0.1856	0.2198	0.1872	0.2299	0.1514
PMU-23	0.1952	0.1770	0.1951	0.1709	0.1904	0.1802	0.1951	0.1770	0.2034	0.1794	0.2066	0.1703
PMU-30	0.1695	0.1197	0.1701	0.0901	0.1629	0.1017	0.1695	0.1196	0.1765	0.1462	0.1801	0.1261
PMU-37	0.1737	0.1244	0.1655	0.1122	0.1692	0.0911	0.1737	0.1315	0.1876	0.1801	0.1740	0.1242
PMU-49	0.1969	0.1048	0.1617	0.1176	0.1950	0.0482	0.1968	0.1048	0.2026	0.1706	0.2061	0.0847
PMU-56	0.2121	0.1797	0.1781	0.1558	0.2131	0.1692	0.2121	0.1797	0.2179	0.1894	0.2213	0.1407
PMU-77	0.2180	0.1446	0.1830	0.1328	0.2120	0.1365	0.2180	0.1445	0.2237	0.1577	0.2275	0.1032
PMU-85	0.2177	0.1225	0.1823	0.1206	0.2108	0.1156	0.2176	0.1225	0.2232	0.1822	0.2271	0.1408
PMU-94	0.2083	0.1055	0.1729	0.0879	0.2014	0.1018	0.2082	0.0869	0.2138	0.1250	0.2177	0.0896
PMU-105	0.2196	0.1115	0.1842	0.0988	0.2127	0.1043	0.2195	0.1150	0.2251	0.1829	0.2290	0.1264
PMU-110	0.2188	0.1370	0.1834	0.1281	0.2119	0.0946	0.2188	0.1369	0.2243	0.1580	0.2282	0.0827

Table 11 contains the results of AVA error for IEEE 118 bus system and the angles are measured in degrees. These results clearly reflect that inclusion of PMUs in the test case of IEEE 118 bus system has reduced the error in state estimation. Both state estimators, WLAV and LMR, have achieved higher estimation accuracy if compared with 'No PMU' case, however, the LMR has much lower values of AVA errors than WLAV in all PMU placements. Any optimal location for PMU placement cannot be advised from these results because optimal placement of PMUs is not the scope of this work and optimal placement problem has its own objective function with defined constraints. IEEE 118 bus is a large power system compared with IEEE 30 bus system and our proposed approach has successfully achieve higher level of accuracy even in large power system.

Table 11. IEEE 118 Bus - Absolute voltage angle (AVA) error.

PMUlocation	White Noise		SBD aspower flowat \(PF_{23-32}\)		SBD as powerinjectionat \(PG_{66}\)		SBD as voltage magnitude at \(Vm _{36}\)		MNIBD at \(PF_{4-11}\), \(PG_{16}\), \(Vm_{3}\)		MNIBD at \(PF_{34-36}\) , \( PG_{33}\), \(Vm_{18}\)
	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR	WLAV	LMR
NO PMU	6.5645	5.7658	8.4241	6.9760	6.9005	6.2859	6.5645	5.7658	8.7292	8.4180	8.8401	7.8542
PMU-5	4.8424	4.1135	5.5419	4.4315	4.1214	4.0842	4.8424	4.1136	7.1270	6.2995	6.6821	6.5527
PMU-12	5.6592	4.1274	4.9530	4.7770	5.0185	4.2916	5.6592	4.1614	7.6667	6.6985	7.7004	7.4777
PMU-23	5.1023	3.9799	5.1023	4.4299	4.4990	4.0794	5.1023	3.9799	7.4183	6.7431	8.1718	6.9019
PMU-37	4.9576	3.8861	6.7908	4.3050	4.5046	4.0361	4.9576	3.8754	6.4610	6.4533	4.2436	3.6609
PMU-49	4.9691	3.3357	4.8502	4.1246	4.6477	4.1873	4.9691	3.3368	7.8704	5.5164	6.6806	6.6307
PMU-56	4.7916	3.6311	6.8126	4.4770	5.0280	4.7945	4.7916	3.6311	7.6820	6.1747	7.4555	6.6350
PMU-77	5.1490	4.1643	7.6845	4.3710	4.8917	4.8848	5.1490	4.1636	7.8193	6.8693	7.9457	7.1038
PMU-85	5.2203	3.6269	7.6517	4.6830	4.8087	4.5386	5.2203	3.6269	7.8273	6.3858	7.5832	7.1190
PMU-94	5.4897	3.7719	7.9211	4.0410	5.0780	4.0052	5.4897	3.4912	8.0966	6.2631	7.3938	7.3884
PMU-105	5.0139	3.9057	7.4453	4.7130	4.6023	4.0723	5.0139	3.9050	7.6209	6.4835	7.5176	6.9126
PMU-110	4.9884	3.8172	7.4198	4.7730	4.5768	4.4168	4.9884	3.8172	7.5954	6.0187	7.6356	6.8871

4. Conclusion and Future Directions

In this paper, the proposed approach to incorporate both voltage and current phasors from PMUs into the existing robust estimators: WLAV and LMR is successfully implemented and evaluated. The use of LMR estimator in the presence of voltage and current phasors coming from different locations is investigated provided significant improvement in the accuracy of the state estimation. The overall performance for LMR estimator compared with WLAV estimator proved to be better irrespective of the bad-data presence and test case size. Besides of the single bad-data cases, multiple interacting and non-interacting bad-data cases were also investigated that clearly reflected the robustness of LMR estimator and better estimation results compared with WLAV estimator. The working principle of LMR estimator depends upon proper tolerance level that assists in rejection of unreliable or corrupted measurement values as explained in the section 2. In this paper, the tolerance value of LMR estimator is tuned iteratively. The accurate meter readings from PMUs were easily incorporated into WLAV estimator by using covariance matrix. But for the LMR estimator, it was not possible to apply such higher weights to PMU meter readings like WLAV. The simulation study concluded into a pertinent contribution that the tolerance should be kept zero for all PMU meter readings. The accuracy of the estimator also depends upon the proper selection of the buses where the new PMUs are to be installed. This issue is carefully addressed in the paper for the test cases of IEEE 30 and IEEE 118 bus system.

Acknowledgements

The authors acknowledge the support of King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia.

Competing Interests

The authors declare no competing interest.

1. Introduction

In Gas Metal Arc Welding (GMAW), materials are melted and joined by heating them by an arc which is produced between filler wire which is continuously fed the metals. Arc is protected and shielded by using inert gases like Helium and Argon and because of this GMAW is also called Metal Inert Gas (MIG) process. Other than inert gases some active gases i.e. \(CO_{2}\) are used so GMAW is now the widely used name for this process. This is constant voltage process and Direct Current Electrode Positive (DCEN) is used in GMAW. Either Direct Current Electrode Positive (DCEN) or Alternating Current (AC) transfer of metal is unreliable. But here arc is a stable transfer of metal smoothly, low chances of spatters, and high penetration is worthy characteristics of this process [1, 2].

M. Azadi [3] studied the optimization of GMAW over API-X42 material through L36 Taguchi matrix and proposed back propagation neural network (BPNN). Omer Bataineh [4] studied the GMAW process to find the effects of welding factors over welding strength; the optimization was done through factorial design methodology and ANOVA. Izzatul [5] optimized the welding responses such as penetration, microstructural and hardness by utilizing robotic GMAW process. Pawan Kumar [6] studied the dissimilar welding joints welded by GMAW and optimized the result by Taguchi orthogonal L9 array and analysis was concluded based on the signal to noise ratios. Chandresh [7] worked with full factorial design experiment to optimize the GMAW responses on AISI 1020 material.

All above investigation and experimental study were conducted on GMAW process under a controlled environment such as well-established and equipped closed welding workshop whereas we have studied the semi-automatic GMAW performed under tough environmental conditions such as windy and dusty in the mobile or outdoor welding shop. Through data mining of field engineering on nominal identical data on which we have done field study and relevant data points are chosen. The portable welding shop is the shop which moves the location to location base on projects scope such as for cross-country pipeline.

2. Process Features and Parameters [1]

In the gas metal welding, arc regulation is auto-adjusted by any of two ways. The most known technique is to employ a constant-speed electrode feed unit with a variable current and constant voltage source. With the variation of gap between gun and work piece, the variation in arc length is also noted, where if arc length decrease power source gives more current and if arc length increases then of course less current. So there will be relevant changes in meting rate due to such changes in current which in turn require maintaining arc length. Other technique of arc regulation is to employ a constant-current source along with adjustable-speed, voltage-sensing electrode feeder. Here any change in arc length originates a relevant variation in the AC voltage. Based on the electrode feed unit changes the speed of filler metal wire depending upon higher/lower speed. This is more useful for larger electrodes when lower feed speed of filler metal wire is required. The GMAW setup used for the research work is depicted in Figure 1.

3. Design of Experiment Methodology [8, 9]

The factors can be classified as either continuous with low and high value or categorical with different level. We have selected continuous type of factors instead of the categorical type with minimum and maximum value because GMAW factor's values fluctuate frequently due to welding in the temporary welding shop. Design of experiment is done by selecting voltage (V), welding consumable feed speed (mm), and travel speed (cm/mm) as continuously controllable factors with low and high values as defined in Table 2. These controllable factor's effects are studied on four responses such as weld bead (mm), bead reinforcement height (mm), penetration depth (mm) and deposition efficiency(%). Each response is assigned with minimum and maximum values which are obtained from design condition for ASME B31.3 process piping for non-sour and non-lethal service and are mentioned in Table 1. The response's results are studied and analyzed on mean values basis and target to maximize the mean of the depth of penetration, deposition efficiency. The quantitative relation between responses and factors in term of Response Surface Methodology (RSM) can be expressed as below: \begin{equation*} Y = f( \text{Voltage, wire feed , travel speed}) \end{equation*} whereas \(Y\) is the responses which are to be optimized and \(V, F\) \& \(S\) are the controllable factors. The system behavior will be obtained through quadric model or by higher order polynomial model which is developed by the least square method by considering the interaction of factors to maximize or minimize the response variables, below is the quadratic model for author case: In Equation (1), the author designate betas as coefficient of linear, quadratic and interaction of input \(V\), \(F\), and \(T\). The \(\beta0\) is the intercept term whereas \(\beta1, \ \beta2, \ \beta3 \ and \ \beta11,\ \beta22, \ \beta33\) are the linear terms and interaction between variables terms respectively.

Table 1. Defining of controllable factors to be measured.

Name	Units	Type	Role	Low	High
A:Welding Voltage	V	Continuous	Controllable	29.0	34.2
B: Wire Feed Speed	mm	Continuous	Controllable	3.9	9.7
C:Welding Speed	mm/min	Continuous	Controllable	50.0	70.0

Table 2. Defining of responses to be varied.

Name	Units	Analyze	Goal	Impact	Sensitivity	Low	High
Depth of Penetration	mm	Mean	Maximize	3.0	Medium	4.5	6.0
Deposition Efficiency	%	Mean	Maximize	3.0	Medium	55.6	93.4
Bead Width	mm	Mean	Maximize	3.0	Medium	6.5	8.0
Bead Reinforcement	mm	Mean	Maximize	3.0	Medium	0.7	3.0

Design of experiment's run and the model was established by Box-Behnken design which was developed by George E. P. Box. Box-Behnken is one of the experimental strategies for RSM where each factor is placed at equally distributed values on at least three levels and it fit on the quadratic model the design is equidistant from the design center which is placed in last. Total 30 runs are designed with 3 center points per block. The Design of experiments (DOE) parameters are tabulated in Table 3 [8].

Table 3. Design of experiments for a gas metal arc welding process.

Type of  Factors	Design Type	Center points Per Block	CenterPoint Placement	Design is Randomized	Number of  Replicates	Total Runs	Total Blocks
Process	Box-Behnken design	3	Last	Yes	1	30	2

To fit the results of the experiment, the Quadratic model of factors interaction is used and below table is the statistical data collected from large data points of the mobile welding shop, 30 runs are collected and responses of each run are measured from actual testing in mechanical laboratory refer to Table 4 for detail. The welding was conducted on API 5L Gr. 70 (fine grain Normalized 19mm thickness) material with single bevel angle. Bevel angle was checked with Dye penetrant testing for any possible defects like crack etc. GMAW machine of model MAXI 505 was used for welding. Whereas reinforcement was measured by using Cambridge type welding gauge. The welding joint design is given below in Figure 2; (joint design and bead width and reinforcement height etc [2].)

The larger pool of field data for GMAW process is reported in Raw Data file by authors [9].

Table 4. GMAW data for optimization process.

Sr. No.	Factors			Responses
	F1	F2	F3	R1	R2	R3	R4
	Voltage	Wire Feed	Travel Speed	Depth of Penetration (mm)	Deposition Efficiency (%)	Bead Width (mm)	Weld Reinforcement (mm)
1	29	6.8	50.0	5	55.6	6.50	0.7
2	34.2	6.8	70.0	6	91.6	6.50	2.0
3	29	9.7	60.0	5	90	6.00	2.0
4	31.6	3.9	50.0	5.5	77	8.50	3.0
5	29	3.9	70.0	5	77	8.00	2.5
6	31.6	3.9	50.0	4.5	56	6.50	0.8
7	34.2	6.8	70.0	5.5	92.5	6.50	2.2
8	34.2	9.7	60.0	6	92	6.00	2.2
9	31.6	9.7	50.0	6.5	80	8.00	3.0
10	31.6	9.7	60.0	5.3	92.3	6.50	2.0
11	34.2	3.9	70.0	5.2	92.1	6.40	2.0
12	29	6.8	50.0	5.3	92.3	6.50	2.0
13	31.6	6.8	60.0	5	92.2	6.50	2.0
14	34.2	9.7	70.0	5.3	88.7	7.0	3.0
15	31.6	3.9	50.0	5.6	90.2	7.7	2.9
16	31.6	6.8	60.0	5.3	90.7	6.3	2.5
17	34.2	3.9	70.0	5.3	88.6	7.8	3.1
18	29	6.8	50.0	6.0	94.7	7.3	2.7
19	31.6	9.7	60.0	6.1	94.0	9.4	2.9
20	34.2	9.7	70.0	6.1	94.6	6.1	2.5
21	31.6	3.9	50.0	5.7	93.0	8.5	2.9
22	31.6	6.8	60.0	5.2	88.3	6.3	2.9
23	34.2	6.8	70.0	5.2	87.7	7.0	3.0
24	29	9.7	50.0	5.2	89.1	6.3	2.5
25	31.6	3.9	60.0	5.4	89.4	7.7	2.9
26	34.2	6.8	70.0	5.8	93.3	7.3	2.7
27	29	3.9	50.0	6.3	96.5	9.6	3.4
28	31.6	6.8	60.0	5.9	95.2	6.6	2.6
29	34.2	9.7	70.0	6.3	98.9	10.5	2.6
30	31.6	3.9	50.0	5.6	91.7	6.6	2.8

4. Analyze the Experiments Results [8]

The variance's analysis is performed by Analysis of variance (ANOVA) was conducted to investigate the controllable factor's influences on measured responses then the standardized Pareto chart is drawn for each response with respect to significant factors. The analysis of variance is calculated for each of response and is explained in detail, refer below to Table 5 for depth of penetration, Table 6 for deposition efficiency, Table 7 for bead width and Table 8 for weld reinforcement height:

Table 5. Variance's analysis for SQRT of Depth of Penetratione.

Source	Sum of Squares	Dof	Mean Square	F-Ratio	P-Value
A:Welding Voltage	0.0330452	1	0.0330452	1.73	0.2799
B:Wire Feed Speed	0.0384051	1	0.0384051	2.01	0.2512
C:Welding Speed	0.0113129	1	0.0113129	0.59	0.4975
AA	0.00000229293	1	0.00000229293	0.00	0.9919
AB	0.00715199	1	0.00715199	0.37	0.5838
AC	0.00036437	1	0.00036437	0.02	0.8989
BB	0.0000164209	1	0.0000164209	0.00	0.9784
BC	0.000137468	1	0.000137468	0.01	0.9377
CC	0.0020654	1	0.0020654	0.11	0.7639
Total error	0.0572911	3	0.019097
Total (corr.)	0.150612	12

Table 5 is the variance analysis known as ANOVA where statistical significance is calculated by associating the experimental error with a mean square. The mean square value was obtained by computing the variability in Sq-Root (DOP) into a distinct run for each of the effects. In author occasion, no interactions of factors have P-values less than 0.05 which confirmed that all interactions are meaningfully diverse from "0" at the 95.0% confidence level.

The R-Squared calculation shows that the model as fitted illuminates 59.7% of the variability in Sq-Root (Depth of Penetration). The adjusted R-squared calculation is 39.55%, The standard error of the calculation indicates as 0.0871038.
The mean absolute error of 0.0590065 is the average value of the residuals.

Based on analysis of variance through ANOVA, the significant controllable factors are identified and then plotted against the standardized effect. Figure 3 shows that Wire Feed, welding voltage and their quadratic effect have a major effect on depth of penetration as compared to other factors. P-value for all interactions are greater than 5.0%.

Table 6. Variance's analysis for SQRT of Deposition Efficiency.

Source	Sum of Squares	Dof	Mean Square	F-Ratio	P-Value
A:Welding Voltage	1.16387	1	1.16387	2.45	0.2158
B:Wire Feed Speed	1.12495	1	1.12495	2.36	0.2217
C:Welding Speed	0.272031	1	0.272031	0.57	0.5045
AA	0.00838265	1	0.00838265	0.02	0.9028
AB	0.128551	1	0.128551	0.27	0.6391
AC	1.20742	1	1.20742	2.54	0.2094
BB	0.205795	1	0.205795	0.43	0.5577
BC	0.95518	1	0.95518	2.01	0.2515
CC	0.821663	1	0.821663	1.73	0.2802
Total error	1.42735	3	0.475783
Total (corr.)	7.44221	12

Table 6 is the variance analysis known as ANOVA where statistical significance is calculated by associating the experimental error with a mean square. The mean square value was obtained by computing the variability in Sq-Root (DOP) into a distinct run for each of the effects. In author occasion, No interactions of factors have P-values less than 0.05 which confirmed that all interactions are meaningfully diverse from “0” at the 95.0% confidence level. The R-Squared calculation shows that the model as fitted illuminates 80.82% of the variability in Sq-Root (Depth of Penetration). The adjusted R-squared calculation is 23.22%, The standard error of the calculation shows 0.689771 for the standard deviation of the residuals. The mean absolute error of 0.271215 is the average value of the residuals.

Based on analysis of variance through ANOVA, the significant controllable factors are identified and then plotted against the standardized effect. Figure 4 shows that quadric effect of Welding voltage and wire feed and wire feed and welding speed have a major effect on deposition efficiency as compared to other factors. P-value for all interactions is greater than 5.0%.

Table 7. Variance's analysis for SQRT of Bead Width.

Source	Sum of Squares	DOF	Mean Square	F-Ratio	P-Value
A:Welding Voltage	0.0111456	1	0.0111456	0.42	0.5650
B:Wire Feed Speed	0.037308	1	0.037308	1.39	0.3232
C:Welding Speed	0.0519844	1	0.0519844	1.94	0.2581
AA	0.0122513	1	0.0122513	0.46	0.5475
AB	0.0222912	1	0.0222912	0.83	0.4291
AC	0.0	1	0.0	0.00	1.0000
BB	0.0177043	1	0.0177043	0.66	0.4760
BC	0.00189437	1	0.00189437	0.07	0.8076
CC	0.0122513	1	0.0122513	0.46	0.5475
Total error	0.0804485	3	0.0268162
Total (corr.)	0.271784	12

The R-Squared calculation shows that the model as fitted illuminates 70.3999% of the variability in Sq-Root (Depth of Penetration). The adjusted R-squared calculation is 0.0%, which is appropriate for analyzing matrix with dissimilar number of independent variables. The estimated standard error for standard deviation of the residuals to be 0.163756. The mean absolute error of 0.0639226is the average value of the residuals.

Based on analysis of variance through ANOVA, the significant controllable factors are identified and then plotted against the standardized effect. Figure 5 shows that Welding speed and wire feed have a major effect on bead width as compared to other factors. There is no figure for serial autocorrelation in the residuals at significance level of 5.0% because P-value for interactions is greater than 5.0%.

Table 8. Variance's analysis for SQRT of Weld Reinforcement.

Source	Sum of Squares	DOF	Mean Square	F-Ratio	P-Value
A:Welding Voltage	0.0376313	1	0.0376313	0.24	0.6554
B:Wire Feed Speed	0.0222486	1	0.0222486	0.14	0.7295
C:Welding Speed	0.0523154	1	0.0523154	0.34	0.6013
AA	0.00538944	1	0.00538944	0.03	0.8637
AB	0.0139183	1	0.0139183	0.09	0.7836
AC	0.104516	1	0.104516	0.68	0.4709
BB	0.0264368	1	0.0264368	0.17	0.7068
BC	0.0675445	1	0.0675445	0.44	0.5556
CC	0.0141117	1	0.0141117	0.09	0.7821
Total error	0.46308	3	0.15436
Total (corr.)	0.841292	12

The R-Squared calculation shows that the model as fitted illuminates 44.9561% of the variability in Sq-Root (Depth of Penetration). The adjusted R-squared (the appropriate one) calculation is 0.0%, .The estimated standard error of the residuals to be 0.392887. The mean absolute error of 0.156633 is the average value of the residuals.

Based on analysis of variance through ANOVA, the significant controllable factors are identified and then plotted against the standardized effect. Figure 6 shows that Quadric effects of Welding voltage and welding speed have a major effect on bead width as compared to other factors.

5. Optimization through Desirability-Function [8]

Optimum factors setting values are obtained through the value of optimized desirability which is obtained by considering the combined results of all five responses for specific predicted values. For optimization of response values, we have selected confident level of 95%. Whereas predicted values of each response is a mean value of upper 95% and lower 95% limit. Observed desirability is calculated from observed values of all response's values for each run of an experiment by using Derringer's model. Whereas predicted desirability is obtained from predicted values of all response base on lower and upper 95% limits of the confidence level.

6. Desirability Function [9, 10, 11 ]

The desirability function and loss function are used optimized approaches and we have used desirability function methodology because it is more applicability and flexibility as compared to loss function moreover it is a most suitable function to solve multi-objective optimization problems. As per definition is given by Harrington [9], "converts every response's value into scale-free value is known as desirability". The desirability function was also studied and shared by Derringer and Suich's [10] which is used for nominal-the best (NTB), larger-the-best (LTB), smaller-the-best (STB) type of measurable responses. In our DOE, we have selected the larger the best which is defined as: \[ d(i)= \begin{cases} 0, & yl \leq LSL_{i} \\ \frac{\hat{y}-USL_{i}}{LSL_{i}-USL_{i}}, & LSL \le \hat{y} \le USLi{i}\\ 1, & yl = USL_{i} \end{cases} \] Where \(d(i)\) is the desirability function, yet is the response, \(USL_{i}\) is the upper 95% limit and \(LSL_{i}\) is the lower 95% limits. Table 9 shows all calculation where responses are optimized by getting their prediction by taking the mean of lower 95.0% limits and upper 95.0% limit. Base on Derringer and Suich's desirability function as defined above, the desirability is calculated for depth of penetration, deposition efficiency, bead width, and bead reinforcement are 52.8%, 85.3%, 81.3% and 93.9% respectively [8]. The overall desirability or optimized desirability is equaled 77%, which is obtained by considering the desirability of all responses then taking each value to the power equal to its impact, taking the product of both results, and the resultant product raises to a power equal to 1 divided by impact summation.

Table 9. Optimum Response Values.

Response	Optimized	Prediction	Lower 95.0% Limit	Upper 95.0% Limit	Desirability
Depth of Penetration	yes	5.2921	3.47671	7.48744	0.528068
Deposition Efficiency	yes	87.8636	51.8112	133.382	0.853534
Bead Width	yes	7.72064	5.11714	10.8577	0.813758
Bead Reinforcement	yes	2.8614	0.204658	8.58927	0.939741

Optimized Desirability = 77%. Optimums setting of factors are obtained based on optimized desirability vs optimized responses values and are given in Table 10 and the graphical representation is mentioned in Figure 7.

Table 10. Factor settings at optimum.

Factor	Setting
Welding Voltage	33.2783
Wire Feed Speed	3.9
Welding Speed	60.0

The graphical presentation of optimal result shows in Figure 7 and Figure 8. Where within the model with respect to controllable welding variables desirability zone are marked in different color, blue color zone depicted the undesirable zone with value of 0 and red zone depict the ideal condition with 100% desirability. Base on welding condition and variables range, optimal desirability zone is 77% for the given values of welding parameters.

7. Validation, Discussion, and Conclusion

After getting the optimal values for factors next stage was to validate these values. In order to do this final weld, the run was conducted by using these optimal values obtained during these analyses. Welding was performed under same circumstances using the same material. Final results were obtained and found very close to the optimal responses as mentioned in Table 11.

Table 11. Comparison between optimized and actual results.

Optimized Factors	Optimized Responses	Actual Results (Responses)
Welding Voltage = 33.2 V	Depth of Penetration = 5.2921 mm	Depth of Penetration = 5.3 mm
Wire Feed Speed = 3. mm	Deposition Efficiency =87.8 %	Deposition Efficiency = 90 %
Welding Speed = 60 cm/ mm	Bead Width = 7.72 mm	Bead Width = 6.52 mm
	Reinforcement = 2.86 mm	Reinforcement = 2.2 mm

This study discloses the application of RSM Box-Behnken design and desirability analyses for gas metal arc welding to optimize the welding output. By using these methodologies optimal values for selected controllable factors and responses are found and then optimal responses values are validated by performing an actual run of the weld and then comparing the actual results with the calculated optimal values. Hence conclusion can be made that for given certain material and similar welding circumstances these analyses can be used for the best quality weld.

Acknowledgements

The completion of this research work could not be possible without support and assistance of so many people whose names may not all be enumerated, however, we would like to express our thanks to particular Mr. Ali Raza lone - QC manager of Saudi Arkad, Mr. Azhar - Ph.D. student of King Fahd University of Petroleum and Minerals (KFUPM), Mr. Saravanan - JGC Department Manager of Construction and the most Mr. ITO Kenji - Senior Manager JGC for their endless support during my research work. The research work's Abstract of same methodology on Flux-cored arc welding (FCAW) was also presented in 15th Annual Congress on Materials Research & Technology [13].

Competing Interests

The authors declare no competing interest.

1. Introduction

Chemical reaction network theory deals with an attempt to model the behavior of real world chemical systems. From the very beginning of its foundation, it is hot cake for research community; especially due to its importance in two important branches i.e. biochemistry and theoretical chemistry. It has also a significant place in pure mathematics particularly due to its mathematical structures.

Cheminformatics is an upcoming and progressive area that deals with the relationships of qualitative structure activity (QSAR) and structure property (QSPR) and also predicts the biochemical activities and properties of nanomaterial. In these studies, for the prediction of bioactivity of the chemical compounds, some physcio-chemical properties and topological indices are used see [1, 2, 3, 4].

Mathematical chemistry is the branch of chemistry which discusses the chemical structures with the aid of mathematical tools. Molecular graph is a simple connected graph in chemical graph theory. This graph consists of atoms and chemical bonds and they are represented by vertices and edges respectively. The distance between two vertices \(u\) and \(v\) is represented as \(d(u,v)\) and it is the shortest length between \(u\) and \(v\) in graph \(G.\) The degree of vertex is basically the number of vertices of \(G\) adjacent to a given vertex \(v\) and will be denoted by \(d_v\).

The topological index of a molecule can be used to quantify the molecular structure. To be simple, the topological index can be considered a function that assign each molecular structure to real number. Boiling point, heat of evaporation, heat of formation, chromatographic retention times, surface tension, vapor pressure etc can be predicted by using topological indices. First and second Zagreb indices are degree based graph invariants have been studies extensively since 1970's.

The first and second K Banhatti indices were introduced by Kulli in [5] as $$B_{1} (G)=\sum _{ue}[d_{G} (u) +d_{G} (e)]$$ and $$B_{2} (G)=\sum _{ue}[d_{G} (u) d_{G} (e)].$$ The first and second multiplicative K Banhatti indices were introduced by Kulli in [6] as $$BII_{1} (G)=\prod _{ue}[d_{G} (u) +d_{G} (e)]$$ and $$BII_{2} (G)=\prod _{ue}[d_{G} (u) d_{G} (e)].$$ The following K hyper-Banahatti indices are defined in [6] as $$HB_{1} (G)=\sum _{ue}[d_{G} (u) +d_{G} (e)]^{2} $$ and $$HB_{2} (G)=\sum _{ue}[d_{G} (u) d_{G} (e)]^{2} .$$ The first and second multiplicative K hyper-Banhatti indices are defined as $$HBII_{1} (G)=\prod _{ue}[d_{G} (u) +d_{G} (e)]^{2} $$ and $$HBII_{2} (G)=\prod _{ue}[d_{G} (u) d_{G} (e)]^{2} .$$ The K harmonic Banhatti index is defined as $$H_{b} (G)=\sum _{ue}[\frac{2}{d_{G} (u)+d_{G} (e)} ] .$$ The multiplicative K harmonic Banhatti index is defined as $$HII_{b} (G)=\prod _{ue}[\frac{2}{d_{G} (u)+d_{G} (e)} ] .$$ In this paper we compute several Banhatti type indices of \(TUC_4[m,n]\), \(TUZC_6[m,n]\), \(TUZC_6[m,n]\), \(SC_5C_7[p,q]\), \(NPHX[p,q]\), \(VC_5C_7[p,q]\) and \(HC_5C_7[p,q]\) nanotubes.

2. Main Results

2.1. Banhatti indices of \(TUC_4[m,n]\)

In the nanoscience, \(TUC_4[m,n]\) nanotubes (where \(m\) and \(n\) are denoted as the number of squares in a row and the number of squares in a column respectively.) are plane tiling of \(C_4\). This tessellation of \(C_4\) can cover either a torus or a cylinder. The 3D representation of \(TUC_4[m,n]\) is described in Figure 1.

Theorem 1. Let \(G\) be the \(TUC_{4} [m,n]\) nanotube. Then we have

1. \(B_{1} (TUC_{4} [m,n])=40mn+2m.\)
2. \(B_{2} (TUC_{4} [m,n])=96mn-26m.\)
3. \(HB_{1} (TUC_{4} [m,n])=400mn-144m.\)
4. \(HB_{2} (TUC_{4} [m,n])=2304mn-1630m\)
5. \(H_{b} (TUC_{4} [m,n])=\frac{4}{5} mn+\frac{559}{630} m.\)

Proof. Let \(G=TUC_{4} [m,n].\) The edge set of \(UC_{4} [m,n]\) can be partitioned as follows:
\(E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\} ,\)
\(E_{7} =\{ uv\in E(G):d_{G} (u)=3,d_{G} (v)=4\},\)
\(E_{8} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=4\},\)
such that \(|E_{6} |=2m,\) \(|E_{7} |=2m\) and \(|E_{8} |=m(2n-3).\)
The edge degree partition of \(V\) is given in Table 1.

Table 1. Edge degree partition of \(TUC_{4} [m,n]\).

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((3,3)\)	\((3,4)\)	\((4,4)\)
\(d_{G} (e)\)	\(4\)	\(5\)	\(6\)
Number of edges	\(2m\)	\(2m\)	\(m(2n-3)\)

Now

1. First K Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} B_{1} (TUC_{4} [m,n])&=&(2m)[(3+4)+(3+4)]+(2m)[(3+5)+(4+5)]+(m(2n-3))[(4+6)+(4+6)]\\ &=&40mn+2m.\end{eqnarray*}
2. Second K Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} B_{2} (TUC_{4} [m,n])&=&(2m)[(3\times 4)+(3\times 4)]+(2m)[(3\times 5)+(4\times 5)]+(m(2n-3))[(4\times 6)+(4\times 6)]\\ &=&96mn-26m.\end{eqnarray*}
3. First K hyper-Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} HB_{1} (TUC_{4} [m,n])&=&(2m)[(3+4)^{2} +(3+4)^{2} ]+(2m)[(3+5)^{2} +(4+5)^{2} ]\\&&+(m(2n-3))[(4+6)^{2} +(4+6)^{2} ]\\ &=&400mn-144m.\end{eqnarray*}
4. Second K hyper-Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} HB_{2} (TUC_{4} [m,n])&=&(2m)[(3\times 4)^{2} +(3\times 4)^{2} ]+(2m)[(3\times 5)^{2} +(4\times 5)^{2} ]\\&&+(m(2n-3))[(4\times 6)^{2} +(4\times 6)^{2}]\\&=&2304mn-1630m.\end{eqnarray*}
5. K Banhatti harmonic index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} H_{b} (TUC_{4} [m,n])&=&(2m)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]+(2m)[(\frac{2}{3+5} )+(\frac{2}{4+5} )]\end{eqnarray*}\begin{eqnarray*}&&+m(2n-3)[(\frac{2}{4+6} )+(\frac{2}{4+6} )]\\ &=&\frac{4}{5} mn+\frac{559}{630} m.\end{eqnarray*}

Theorem 2. Let \(G\) be the \(TUC_{4} [m,n]\) nanotube. Then we have

1. \(BII_{1} (TUC_{4} [m,n])={ 7}^{4m} \times 8^{2m} \times 9^{2m} \times 10^{2m(2n-3)} .\)
2. \(BII_{2} (TUC_{4} [m,n])={ 12}^{4m} \times 15^{2m} \times 20^{2m} \times 24^{2m(2n-3)}. \)
3. \(HBII_{1} (TUC_{4} [m,n])={ 7}^{8m} \times 8^{4m} \times 9^{4m} \times 10^{4m(2n-3)} .\)
4. \(HBII_{2} (TUC_{4} [m,n])={ 12}^{8m} \times 15^{4m} \times 20^{4m} \times 24^{4m(2n-3)}. \)
5. \(HII_{b} (TUC_{4} [m,n])=(\frac{2}{7} )^{4m} \times (\frac{1}{4} )^{2m} \times (\frac{2}{9} )^{2m} \times (\frac{1}{5} )^{2m(2n-3)} .\)

Proof.

1. First multiplicative K Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} BII_{1} (TUC_{4} [m,n])&=&[(3+4)^{(2m)} \times (3+4)^{(2m)} ]\times [(3+5)^{(2m)} \times (4+5)^{(2m)} ]\\ && \times [(4+6)^{(m(2n-3))} \times (4+6)^{(m(2n-3))} ]\\ &=& 7^{4m} \times 8^{2m} \times 9^{2m} \times 10^{2m(2n-3)}. \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} BII_{2} (TUC_{4} [m,n])&=&[(3\times 4)^{(2m)} \times (3\times 4)^{(2m)} ]\times [(3\times 5)^{(2m)} \times (4\times 5)^{(2m)} ]\\ &&\times [(4\times 6)^{(m(2n-3))} \times (4\times 6)^{(m(2n-3))} ]\\ &=& 12^{4m} \times 15^{2m} \times 20^{2m} \times 24^{2m(2n-3)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} HBII_{1} (TUC_{4} [m,n])&=&[((3+4)^{2} )^{(2m)} \times ((3+4)^{2} )^{(2m)} ]\times [((3+5)^{2} )^{(2m)} \times ((4+5)^{2} )^{(2m)} ]\\ &&\times [((4+6)^{2} )^{(m(2n-3))} \times ((4+6)^{2} )^{(m(2n-3))} ]\\ &=& 7^{8m} \times 8^{4m} \times 9^{4m} \times 10^{4m(2n-3)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} HBII_{2} (TUC_{4} [m,n])&=&[((3\times 4)^{2} )^{(2m)} \times ((3\times 4)^{2} )^{(2m)} ]\\ &&\times [((3\times 5)^{2} )^{(2m)} \times ((4\times 5)^{2} )^{(2m)} ]\\ &&\times [((4\times 6)^{2} )^{(m(2n-3))} \times ((4\times 6)^{2} )^{(m(2n-3))} ]\\ &=& 12^{8m} \times 15^{4m} \times 20^{4m} \times 24^{4m(2n-3)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(TUC_{4} [m,n]\) is \begin{eqnarray*} HII_{b} (TUC_{4} [m,n])&=&[(\frac{2}{3+4} )^{(2m)} \times (\frac{2}{3+4})^{(2m)} ]\\ &&\times [(\frac{2}{3+5} )^{(2m)} \times (\frac{2}{4+5} )^{(2m)} ]\\ &&\times [(\frac{2}{4+6} )^{m(2n-3)} \times (\frac{2}{4+6} )^{m(2n-3)} ]\\ &=&(\frac{2}{7} )^{4m} \times (\frac{1}{4} )^{2m} \times (\frac{2}{9} )^{2m} \times (\frac{1}{5} )^{2m(2n-3)} . \end{eqnarray*}

2.2. Banhatti indices of \(TUZC_6[m,n]\)

The zigzag nanotube \(TUZC_6[m,n]\), where \(m\) is the number of hexagons in the first row and \(n\) is the number of hexagons in the first column. The molecular structures of \(TUZC_6[m,n]\) can be referred to Figure 2.

Theorem 3. Let \(G\) be the zigzag nanotube \(TUZC_{6} [m,n]\). Then we have

1. \(B_{1} (TUZC_{6} [m,n])=42mn+16m.\)
2. \(B_{2} (TUZC_{6} [m,n])=72mn+12m.\)
3. \(HB_{1} (TUZC_{6} [m,n])=294mn+48m.\)
4. \(HB_{2} (TUZC_{6} [m,n])=864mn-108m.\)
5. \(H_{b} (TUZC_{6} [m,n])=\frac{12}{7} mn-\frac{188}{108} m.\)

Proof. Let \(G=TUZC_{6} [m,n].\) The edge set of \(TUZC_{6} [m,n]\) can be divided into following classes:
\(E_{5} =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\},\)
\(E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\},\)
such that \(|E_{5} |=4m\) and \(|E_{6} |=3mn-2m.\)
The edge degree partition is given in Table 2.

Table 2 . Edge degree partition of \(TUZC_{6} [m,n]\)..

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(4\)	\(3\)
Number of edges	\(3mn-2m\)	\(4m\)

Now

1. First K Banhatti index of \(TUZC_{6} [m,n]\) is \begin{eqnarray*} B_{1} (TUZC_{6} [m,n])&=&(3mn-2m)[(3+4)+(3+4)]+(4m)[(2+3)+(3+3)]\\ &=&42mn+16m. \end{eqnarray*}
2. Second K Banhatti index of \(TUZC_{6} [m,n]\) is \begin{eqnarray*} B_{2} (TUZC_{6} [m,n])&=&(3mn-2m)[(3\times 4)+(3\times 4)]+(4m)[(2\times 3)+(3\times 3)]\\ &=&72mn+12m. \end{eqnarray*}
3. First K hyper-Banhatti index of \(TUZC_{6} [m,n]\) is \begin{eqnarray*} HB_{1} (TUZC_{6} [m,n])&=&(3mn-2m)[(3+4)^{2} +(3+4)^{2} ]+(4m)[(2+3)^{2} +(3+3)^{2} ]\\ &=&294mn+48m. \end{eqnarray*}
4. Second K hyper-Banhatti index of \(TUZC_{6} [m,n]\) is \begin{eqnarray*} HB_{2} (TUZC_{6} [m,n])&=&(3mn-2m)[(3\times 4)^{2} +(3\times 4)^{2} ]+(4m)[(2\times 3)^{2} +(3\times 3)^{2} ]\\ &=&864mn-108m. \end{eqnarray*}
5. K harmonic Banhatti index of \(TUZC_{6} [m,n]\) is \begin{eqnarray*} H_{b} (TUZC_{6} [m,n])&=&(3mn-2m)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]+(4m)[(\frac{2}{2+3} )+(\frac{2}{3+3} )]\\ &=&\frac{12}{7} mn-\frac{188}{108} m. \end{eqnarray*}

Theorem 4. Let \(G\) be the zigzag nanotube \(TUZC_{6} [m,n]\). Then we have

1. \(BII_{1} (TUZC_{6} [m,n])=5^{4m} \times 6^{4m} \times 7^{2m(3n-2)} .\)
2. \(BII_{2} (TUZC_{6} [m,n])=3^{8m} \times 6^{4m} \times 12^{2m(3n-2)} .\)
3. \(HBII_{1} (TUZC_{6} [m,n])=5^{8m} \times 6^{8m} \times 7^{4m(3n-2)} .\)
4. \(HBII_{2} (TUZC_{6} [m,n])=3^{16m} \times 6^{8m} \times 12^{4m(3n-2)}. \)
5. \(HII_{b} (TUZC_{6} [m,n])={ (}\frac{1}{3} )^{4m} \times (\frac{2}{5} )^{4m} \times (\frac{2}{7} )^{2m(3n-2)} .\)

Proof.

1. First multiplicative K Banhatti index of \(TUZC_{6}[m,n]\) is \begin{eqnarray*} BII_{1} (TUZC_{6} [m,n])&=&[(3+4)^{(3mn-2m)} \times (3+4)^{(3mn-2m)} ]\times [(2+3)^{(4m)} \times (3+3)^{(4m)} ]\\ &=&5^{4m} \times 6^{4m} \times 7^{2m(3n-2)} . \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(TUZC_{6}[m,n]\) is \begin{eqnarray*} BII_{2} (TUZC_{6} [m,n])&=&[(3\times 4)^{(3mn-2m)} \times (3\times 4)^{(3mn-2m)} ]\times [(2\times 3)^{(4m)} \times (3\times 3)^{(4m)} ]\\ &=&3^{8m} \times 6^{4m} \times 12^{2m(3n-2)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(TUZC_{6}[m,n]\) is \begin{eqnarray*} HBII_{1} (TUZC_{6} [m,n])&=&[((3+4)^{2} )^{(3mn-2m)} \times ((3+4)^{2} )^{(3mn-2m)} ]\\&&\times [((2+3)^{2} )^{(4m)} \times ((3+3)^{2} )^{(4m)} ]\\ &=&5^{8m} \times 6^{8m} \times 7^{4m(3n-2)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(TUZC_{6}[m,n]\) is \begin{eqnarray*} HBII_{2} (TUZC_{6} [m,n])&=&[((3\times 4)^{2} )^{(3mn-2m)} \times ((3\times 4)^{2} )^{(3mn-2m)} ]\\ &&\times [((2\times 3)^{2} )^{(4m)} \times ((3\times 3)^{2} )^{(4m)} ]\\ &=&3^{16m} \times 6^{8m} \times 12^{4m(3n-2)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(TUZC_{6}[m,n]\) is \begin{eqnarray*} HII_{b} (TUZC_{6} [m,n])&=&[(\frac{2}{3+4} )^{(3mn-2m)} \times (\frac{2}{3+4} )^{(3mn-2m)} ]\times [(\frac{2}{2+3} )^{(4m)} \times (\frac{2}{3+3} )^{(4m)} ]\\ & =&(\frac{1}{3} )^{4m} \times (\frac{2}{5} )^{4m} \times (\frac{2}{7} )^{2m(3n-2)} .\end{eqnarray*}

2.3. Banhatti indices of \(TUAC_6[m,n]\)

The armchair nanotube \(TUAC_6[m,n]\), where \(m\) is the number of hexagons in the first row and \(n\) is the number of hexagons in the first column. The molecular structures of \(TUAC_6[m,n]\) can be referred to Figure 3.

Theorem 5. Let \(G\) be the armchair nanotube \(TUAC_{6} [m,n]\). Then we have

1. \(B_{1} (TUAC_{6} [m,n])=42mn+16m.\)
2. \(B_{2} (TUAC_{6} [m,n])=72mn+14m.\)
3. \(HB_{1} (TUAC_{6} [m,n])=294mn+56m.\)
4. \(HB_{2} (TUAC_{6} [m,n])=864mn-22m.\)
5. \(H_{b} (TUAC_{6} [m,n])=\frac{12}{7} mn+\frac{199}{105} m.\)

Proof. Let \(G=TUAC_{6} [m,n].\) we have edge set of \(TUAC_{6} [m,n]\) can be partitioned as follows:
\(E_{4} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=2\},\)
\(E_{5} =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\},\)
\(E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\} ,\) such that \(|E_{4} |=m,\) \(|E_{5} |=2m\) and \(|E_{6} |=3mn-m.\)
The edge degree partition is given in Table 3.

Table 3. Edge degree partition of \(TUAC_{6} [m,n]\)

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((2,2)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(2\)	\(4\)	\(3\)
Number of edges	\(m\)	\(3mn-m\)	\(2m\)

Now

1. First K Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} B_{1} (TUAC_{6} [m,n])&=&(m)[(2+2)+(2+2)]+(3mn-m)[(3+4)+(3+4)]\\ &&+(2m)[(2+3)+(3+3)]\\ &=&42mn+16m. \end{eqnarray*}
2. Second K Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} B_{2} (TUAC_{6} [m,n])&=&(m)[(2\times 2)+(2\times 2)]+(3mn-m)[(3\times 4)+(3\times 4)]\\ &&+(2m)[(2\times 3)+(3\times 3)]\\ &=&72mn+14m. \end{eqnarray*}
3. First K hyper-Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} HB_{1} (TUAC_{6} [m,n])&=&(m)[(2+2)^{2} +(2+2)^{2} ]+(3mn-m)[(3+4)^{2} +(3+4)^{2} ]\\ &&+(2m)[(2+3)^{2} +(3+3)^{2} ]\\ &=&294mn+56m. \end{eqnarray*}
4. Second K hyper-Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} HB_{2} (TUAC_{6} [m,n])&=&(m)[(2\times 2)^{2} +(2\times 2)^{2} ]+(3mn-m)[(3\times 4)^{2} +(3\times 4)^{2} ]\\&&+(2m)[(2\times 3)^{2} +(3\times 3)^{2} ]\\ &=&864mn-22m. \end{eqnarray*}
5. K harmonic Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} H_{b} (TUAC_{6} [m,n])&=&(m)[(\frac{2}{2+2} )+(\frac{2}{2+2} )]+(3mn-m)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]\\ &&+(2m)[(\frac{2}{2+3} )+(\frac{2}{3+3} )]\\ &=&\frac{12}{7} mn+\frac{199}{105} m. \end{eqnarray*}

Theorem 6. Let \(G\) be the armchair nanotube \(TUAC_{6} [m,n]\). Then we have

1. \(BII_{1} (TUAC_{6} [m,n])=2^{4m} \times 5^{2m} \times 6^{2m} \times 7^{2m(3n-1)} .\)
2. \(BII_{2} (TUAC_{6} [m,n])=2^{4m} \times 3^{4m} \times 6^{2m} \times 12^{2m(3n-1)} .\)
3. \(HBII_{1} (TUAC_{6} [m,n])=2^{8m} \times 5^{4m} \times 6^{4m} \times 7^{4m(3n-1)} .\)
4. \(HBII_{2} (TUAC_{6} [m,n])=2^{8m} \times 3^{8m} \times 6^{4m} \times 7^{4m(3n-1)} .\)
5. \(HII_{b} (TUAC_{6} [m,n])=(\frac{1}{2} )^{2m} \times (\frac{1}{3} )^{2m} \times (\frac{2}{5} )^{2m} \times (\frac{2}{7} )^{2m(3n-1)} .\)

Proof.

1. First multiplicative K Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} BII_{1} (TUAC_{6} [m,n])&=&[(2+2)^{(m)} \times (2+2)^{(m)} ]\times [(3+4)^{(3mn-m)} \times (3+4)^{(3mn-m)} ]\\ &&\times [(2+3)^{(2m)} \times (3+3)^{(2m)} ]\\ &=&2^{4m} \times 5^{2m} \times 6^{2m} \times 7^{2m(3n-1)} . \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} BII_{1} (TUAC_{6} [m,n])&=&[(2\times 2)^{(m)} \times (2\times 2)^{(m)} ]\times [(3\times 4)^{(3mn-m)} \times (3\times 4)^{(3mn-m)} ]\\ &&\times [(2\times 3)^{(2m)} \times (3\times 3)^{(2m)} ]\\ &=&2^{4m} \times 3^{4m} \times 6^{2m} \times 12^{2m(3n-1)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} HBII_{1} (TUAC_{6} [m,n])&=&[((2+2)^{2} )^{(m)} \times ((2+2)^{2} )^{(m)} ]\\ &&\times [((3+4)^{2} )^{(3mn-m)} \times ((3+4)^{2} )^{(3mn-m)} ]\\ &&\times [((2+3)^{2} )^{(2m)} \times ((3+3)^{2} )^{(2m)} ]\\ &=&2^{8m} \times 5^{4m} \times 6^{4m} \times 7^{4m(3n-1)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} HBII_{2} (TUAC_{6} [m,n])&=&[((2\times 2)^{2} )^{(m)} \times ((2\times 2)^{2} )^{(m)} ]\\ &&\times [((3\times 4)^{2} )^{(3mn-m)} \times ((3\times 4)^{2} )^{(3mn-m)} ]\\ &&\times [((2\times 3)^{2} )^{(2m)} \times ((3\times 3)^{2} )^{(2m)} ]\\ &=&2^{8m} \times 3^{8m} \times 6^{4m} \times 12^{4m(3n-1)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(TUAC_{6} [m,n]\) is \begin{eqnarray*} HII_{b} (TUAC_{6} [m,n])&=&[(\frac{2}{2+2} )^{(m)} \times (\frac{2}{2+2} )^{(m)} ]\times [(\frac{2}{3+4} )^{(3mn-m)} \times (\frac{2}{3+4} )^{(3mn-m)} ]\\ &&\times [(\frac{2}{2+3} )^{(2m)} \times (\frac{2}{3+3} )^{(2m)} ]\\ &=&(\frac{1}{2} )^{2m} \times (\frac{1}{3} )^{2m} \times (\frac{2}{5} )^{2m} \times (\frac{2}{7} )^{2m(3n-1)} . \end{eqnarray*}

2.4. Banhatti indices of \(NPHX[p,q]\)

H-Naphtalenic nanotubes \(NPHX[p, q]\) (where \(p\) and \(q\) are denoted as the number of pairs of hexagons in first row and the number of alternative hexagons in a column, respectively) are a trivalent decoration with sequence of \(C_6,\) \(C_6,\) \(C_4,\) \(C_6,\) \(C_6,\) \(C_4,\ldots\) in the first row and a sequence of \(C_6,\) \(C_8,\) \(C_6,\) \(C_8,\ldots\) in the other rows. In other words, this nanolattice can be considered as a plane tiling of \(C_4,\) \(C_6,\) and \(C_8.\) Therefore, this class of tiling can cover either a cylinder or a torus 4.

Theorem 7. Let \(G\) be the H-Naphtalenic nanotube \(NPHX[m,n]\). Then we have

1. \(B_{1} (NPHX[m,n])=210mn-52m.\)
2. \(B_{2} (NPHX[m,n])=360mn-120m.\)
3. \(HB_{1} (NPHX[m,n])=1470mn-492m.\)
4. \(HB_{2} (NPHX[m,n])=4320mn-1944m.\)
5. \(H_{b} (NPHX[m,n])=\frac{60}{7} mn-\frac{33}{7} m.\)

Proof. Let \(G=NPHX[m,n],\) then we have edge division of edge set \(E(NPHX[m,n])\) as follows:
\(E_{5} =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\},\)
\( E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\}\),
such that \(|E_{5} |=8m\) and \(|E_{6} |=15mn-10m.\)
The edge degree partition is given in Table 4.

Table 4. Edge Edge degree partition of \(NPHX[m,n]\).

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(4\)	\(3\)
Number of edges	\(15mn-10m\)	\(8m\)

Now

1. First K Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} B_{1} (NPHX[m,n])&=&(15mn-10m)[(3+4)+(3+4)]+(8m)[(2+3)+(3+3)]\\ &=&210mn-52m.\end{eqnarray*}
2. Second K Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} B_{2} (NPHX[m,n])&=&(15mn-10m)[(3\times 4)+(3\times 4)]+(8m)[(2\times 3)+(3\times 3)]\\ &=&360mn-120m.\end{eqnarray*}
3. First K hyper-Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} HB_{1} (NPHX[m,n])&=&(15mn-10m)[(3+4)^{2} +(3+4)^{2} ]+(8m)[(2+3)^{2} +(3+3)^{2} ]\\ &=&1470mn-492m.\end{eqnarray*}
4. Second K hyper-Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} HB_{2} (NPHX[m,n])&=&(15mn-10m)[(3\times 4)^{2} +(3\times 4)^{2} ]+(8m)[(2\times 3)^{2} +(3\times 3)^{2} ]\\ &=&4320mn-1944m.\end{eqnarray*}
5. K harmonic Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} H_{b} (NPHX[m,n])&=&(15mn-10m)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]+(8m)[(\frac{2}{2+3} )+(\frac{2}{3+3} )]\\ &=&\frac{60}{7} mn-\frac{33}{7} m.\end{eqnarray*}

Theorem 8. Let \(G\) be the H-Naphtalenic nanotube \(NPHX[m,n]\). Then we have

1. \(BII_{1} (NPHX[m,n])={ 5}^{8m} \times 6^{8m} \times 7^{10m(3n-2)}. \)
2. \(BII_{2} (NPHX[m,n])={ 6}^{8m} \times 9^{8m} \times 12^{10m(3n-2)} .\)
3. \(HBII_{1} (NPHX[m,n])={ 5}^{16m} \times 6^{16m} \times 7^{20m(3n-2)} .\)
4. \(HBII_{2} (NPHX[m,n])={ 6}^{16m} \times 9^{16m} \times 12^{20m(3n-2)} .\)
5. \(HII_{b} (NPHX[m,n])=(\frac{2}{5} )^{8m} \times (\frac{1}{3} )^{8m} \times (\frac{2}{7} )^{10m(3n-2)} .\)

Proof.

1. First multiplicative K Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} BII_{1}(NPHX[m,n])&=&[(3+4)^{(15mn-10m)} \times (3+4)^{(15mn-10m)} ]\times [(2+3)^{(8m)} \times (3+3)^{(8m)} ]\\ &=&5^{8m} \times 6^{8m} \times 7^{10m(3n-2)}. \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} BII_{2} (NPHX[m,n])&=&[(3\times 4)^{(15mn-10m)} \times (3\times 4)^{(15mn-10m)} ]\times [(2\times 3)^{(8m)} \times (3\times 3)^{(8m)} ]\\ &=&6^{8m} \times 9^{8m} \times 12^{10m(3n-2)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} HBII_{1} (NPHX[m,n])&=&[((3+4)^{2} )^{(15mn-10m)} \times ((3+4)^{2} )^{(15mn-10m)} ]\\&&\times [((2+3)^{2} )^{(8m)} \times ((3+3)^{2} )^{(8m)} ]\\ &=&5^{16m} \times 6^{16m} \times 7^{20m(3n-2)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} HBII_{2} (NPHX[m,n])&=&[((3+4)^{2} )^{(15mn-10m)} \times ((3+4)^{2} )^{(15mn-10m)} ]\\&&\times [((2+3)^{2} )^{(8m)} \times ((3+3)^{2} )^{(8m)} ]\\ &=&6^{16m} \times 9^{16m} \times 12^{20m(3n-2)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(NPHX[m,n]\) is \begin{eqnarray*} HII_{b} (NPHX[m,n])&=&[(\frac{2}{3+4} )^{(15mn-10m)} \times (\frac{2}{3+4} )^{(15mn-10m)} ]\\&&\times [(\frac{2}{2+3} )^{(8m)} \times (\frac{2}{3+3} )^{(8m)} ]\\ &=&(\frac{2}{5} )^{8m} \times (\frac{1}{3} )^{8m} \times (\frac{2}{7} )^{10m(3n-2)} . \end{eqnarray*}

2.5. Banhatti indices of \(SC_5C_7[p,q]\)

In nanoscience, \(SC_5C_7[p, q]\) (where \(p\) and \(q\) express the number of heptagons in each row and the number of periods in whole lattice respectively) nanotube is a class of \(C_5C_7\)-net which is yielded by alternating \(C_5\) and \(C_7.\) The standard tiling of \(C_5\) and \(C_7\) can cover either a cylinder or a torus and each period of \(SC_5C_7[p, q]\) consisted of three rows (more details on pth period can be referred to in Figure 5.

Theorem 9. Let \(G\) be the \(SC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(B_{1} (SC_{5} C_{7} [p,q])=168pq-52p.\)
2. \(B_{2} (SC_{5} C_{7} [p,q])=288pq-118p.\)
3. \(HB_{1} (SC_{5} C_{7} [p,q])=1176pq-484p.\)
4. \(HB_{2} (SC_{5} C_{7} [p,q])=3456pq-1858p.\)
5. \(H_{b} (SC_{5} C_{7} [p,q])=\frac{48}{7} pq+\frac{9}{35} p.\)

Proof. Let \(G=SC_{5} C_{7} [p,q].\) There are following three types of edges of \(SC_{5} C_{7} [p,q]\), based on the degree of end vertices \(E_{4}(G) =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=2\} ,\)
\(E_{5}(G) =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\} ,\)
\(E_{6}(G) =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\} ,\)
such that
\(|E_{4}(G)|=p\),\(|E_{5}(G)|=6p\) and \(|E_{6}(G)|=12pq-9p.\) The edge degree partition is given in Table 5.

Table 5. Edge degree partition of \(SC_{5} C_{7} [p,q]\)

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((2,2)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(2\)	\(4\)	\(3\)
Number of edges	\(p\)	\(12pq-9p\)	\(6p\)

Now

1. First K Banhatti index of \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*}B_{1} (SC_{5} C_{7} [p,q])&=&(p)[(2+2)+(2+2)]+(12pq-9p)[(3+4)+(3+4)]+(6p)[(2+3)+(3+3)] \\ &=&168pq-52p. \end{eqnarray*}
2. Second K Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*}B_{2} (SC_{5} C_{7} [p,q])&=&(p)[(2\times 2)+(2\times 2)]+(12pq-9p)[(3\times 4)+(3\times 4)]+(6p)[(2\times 3)+(3\times 3)] \\&=&288pq-118p. \end{eqnarray*}
3. First K hyper-Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{1} (SC_{5} C_{7} [p,q])&=&(p)[(2+2)^{2} +(2+2)^{2} ]+(12pq-9p)[(3+4)^{2} +(3+4)^{2} ]\\&&+(6p)[(2+3)^{2} +(3+3)^{2} ] \\&=&1176pq-484p. \end{eqnarray*}
4. Second K hyper-Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{2} (SC_{5} C_{7} [p,q])&=&(p)[(2\times 2)^{2} +(2\times 2)^{2} ]+(12pq-9p)[(3\times 4)^{2} +(3\times 4)^{2} ]\\&&+(6p)[(2\times 3)^{2} +(3\times 3)^{2} ] \\ &=&3456pq-1858p. \end{eqnarray*}
5. K harmonic Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*} H_{b} (SC_{5} C_{7} [p,q])&=&(p)[(\frac{2}{2+2} )+(\frac{2}{2+2} )]+(12pq-9p)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]\\&&+(6p)[(\frac{2}{2+3} )+(\frac{2}{3+3} )] \\ &=&\frac{48}{7} pq+\frac{9}{35} p. \end{eqnarray*}

Theorem 10. Let \(G\) be the \(SC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(BII_{1} (SC_{5} C_{7} [p,q])=4^{2p} \times 5^{p} \times 6^{p} \times 7^{6p(4q-3)} .\)
2. \(BII_{2} (SC_{5} C_{7} [p,q])=4^{2p} \times 6^{p} \times 9^{p} \times 12^{6p(4q-3)} .\)
3. \(HBII_{1} (SC_{5} C_{7} [p,q])=4^{4p} \times 5^{2p} \times 6^{2p} \times 7^{12p(4q-3)}. \)
4. \(HBII_{2} (SC_{5} C_{7} [p,q])=4^{4p} \times 6^{2p} \times 9^{2p} \times 12^{12p(4q-3)}. \)
5. \(HII_{b} (SC_{5} C_{7} [p,q])=(\frac{1}{2} )^{2p} \times (\frac{2}{5} )^{p} \times (\frac{1}{3} )^{p} \times (\frac{2}{7} )^{6p(4q-3)}.\)

Proof. Using Table 5, we have

1. First multiplicative K Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*}BII_{1} (SC_{5} C_{7} [p,q])&=&[(2+2)^{(p)} \times (2+2)^{(p)} ]\times [(3+4)^{(12pq-9p)} \times (3+4)^{(12pq-9p)} ]\\&&\times [(2+3)^{(6p)} \times (3+3)^{(6p)} ] \\ &=&4^{2p} \times 5^{p} \times 6^{p} \times 7^{6p(4q-3)}. \end{eqnarray*}
2. Second multiplicative K Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*}BII_{2} (SC_{5} C_{7} [p,q])&=&[(2\times 2)^{(p)} \times (2\times 2)^{(p)} ]\times [(3\times 4)^{(12pq-9p)} \times (3\times 4)^{(12pq-9p)} ]\\&&\times [(2\times 3)^{(6p)} \times (3\times 3)^{(6p)} ] \\ &=&4^{2p} \times 6^{p} \times 9^{p} \times 12^{6p(4q-3)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*}HBII_{1} (SC_{5} C_{7} [p,q])&=&[((2+2)^{2} )^{(p)} \times ((2+2)^{2} )^{(p)} ]\times [((3+4)^{2} )^{(12pq-9p)} \times ((3+4)^{2} )^{(12pq-9p)} ] \\&&\times [((2+3)^{2} )^{(6p)} \times ((3+3)^{2} )^{(6p)} ] \\ &=&4^{4p} \times 5^{2p} \times 6^{2p} \times 7^{12p(4q-3)} .\end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HBII_{2} (SC_{5} C_{7} [p,q])&=&[((2\times 2)^{2} )^{(p)} \times ((2\times 2)^{2} )^{(p)} ]\times [((3\times 4)^{2} )^{(12pq-9p)} \times ((3\times 4)^{2} )^{(12pq-9p)} ] \\ &&\times [((2\times 3)^{2} )^{(6p)} \times ((3\times 3)^{2} )^{(6p)} ] \\ &=&4^{4p} \times 6^{2p} \times 9^{2p} \times 12^{12p(4q-3)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index \(SC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HII_{b} (SC_{5} C_{7} [p,q])&=&[(\frac{2}{2+2} )^{(p)} \, \times (\frac{2s}{2+2} )^{(p)} ]\times [(\frac{2}{3+4} )^{(12pq-9p)} \times (\frac{2}{3+4} )^{(12pq-9p)} ]\\&&\times [(\frac{2}{2+3} )^{(6p)} \times (\frac{2}{3+3} )^{(6p)} ] \\ &=&(\frac{1}{2} )^{2p} \times (\frac{2}{5} )^{p} \times (\frac{1}{3} )^{p} \times (\frac{2}{7} )^{6p(4q-3)} . \end{eqnarray*}

2.6. Banhatti indices of \(VC_5C_7[p,q]\)

The molecular graphs of carbon nanotubes \(VC_5C_7[p, q]\) is shown in Figure 6. The structures of this nanotubes consist of cycles \(C_5\) and \(C_7\) (\(C_5C_7\) net which is a trivalent decoration constructed by alternating \(C_5\) and \(C_7\)) by different compound. It can cover either a cylinder or a torus.

Theorem 11. Let \(G\) be the \(VC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(B_{1} (VC_{5} C_{7} [p,q])=336pq+48p.\)
2. \(B_{2} (VC_{5} C_{7} [p,q])=576pq+36p.\)
3. \(HB_{1} (VC_{5} C_{7} [p,q])=2352pq+144p.\)
4. \(HB_{2} (VC_{5} C_{7} [p,q])=6912pq-324p.\)
5. \(H_{b} (VC_{5} C_{7} [p,q])=\frac{96}{7} pq+\frac{188}{35} p.\)

Proof. Let \(G=VC_{5} C_{7} [p,q].\) Then the edge set of \(VC_{5} C_{7} [p,q]\) can be partitioned into following two classes:
\(E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\},\)
\(E_{5} =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\},\)
such that \(|E_{6} |=24pq-6p\) and \(|E_{5} |=12p.\)
The edge degree partition is given in Table 6.

Table 6. Edge degree partition of \(VC_{5} C_{7} [p,q]\)

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(4\)	\(3\)
Number of edges	\(12pq-6p\)	\(12p\)

Now

1. First K Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} B_{1} (VC_{5} C_{7} [p,q])&=&(24pq-6p)[(3+4)+(3+4)]+(12p)[(2+3)+(3+3)]\\ &=&336pq+48p. \end{eqnarray*}
2. Second K Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} B_{2} (VC_{5} C_{7} [p,q])&=&(24pq-6p)[(3\times 4)+(3\times 4)]+(12p)[(2\times 3)+(3\times 3)]\\ &=&576pq+36p. \end{eqnarray*}
3. First K hyper-Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{1} (VC_{5} C_{7} [p,q])&=&(24pq-6p)[(3+4)^{2} +(3+4)^{2} ]+(12p)[(2+3)^{2} +(3+3)^{2} ]\\ &=&2352pq+144p. \end{eqnarray*}
4. Second K hyper-Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{2} (VC_{5} C_{7} [p,q])&=&(24pq-6p)[(3\times 4)^{2} +(3\times 4)^{2} ]+(12p)[(2\times 3)^{2} +(3\times 3)^{2} ]\\ &=&6912pq-324p. \end{eqnarray*}
5. K harmonic Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} H_{b} (VC_{5} C_{7} [p,q])&=&(24pq-6p)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]+(12p)[(\frac{2}{2+3} )+(\frac{2}{3+3} )]\\ &=&\frac{96}{7} pq+\frac{188}{35} p. \end{eqnarray*}

Theorem 12. Let \(G\) be the \(VC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(BII_{1} (VC_{5} C_{7} [p,q])=5^{12p} \times 6^{12p} \times 7^{12p(4q-1)} .\)
2. \(BII_{2} (VC_{5} C_{7} [p,q])=3^{12p} \times 6^{12p} \times 12^{12p(4q-1)} .\)
3. \(HBII_{1} (VC_{5} C_{7} [p,q])=5^{24p} \times 6^{24p} \times 7^{24p(4q-1)} .\)
4. \(HBII_{2} (VC_{5} C_{7} [p,q])=3^{24p} \times 6^{24p} \times 12^{24p(4q-1)} .\)
5. \(HII_{b} (VC_{5} C_{7} [p,q])=(\frac{2}{7} )^{12p(4q-1)} \times (\frac{2}{5} )^{12p} \times (\frac{1}{3} )^{12p} .\)

Proof.

1. First multiplicative K Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} BII_{1} (VC_{5} C_{7} [p,q])&=&[(3+4)^{(24pq-6p)} \times (3+4)^{(24pq-6p)} ]\times [(2+3)^{(12p)} \times (3+3)^{(12p)} ]\\ &=&5^{12p} \times 6^{12p} \times 7^{12p(4q-1)} . \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} BII_{2} (VC_{5} C_{7} [p,q])&=&[(3\times 4)^{(24pq-6p)} \times (3\times 4)^{(24pq-6p)} ]\times [(2\times 3)^{(12p)} \times (3\times 3)^{(12p)} ]\\ &=&3^{12p} \times 6^{12p} \times 12^{12p(4q-1)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HBII_{1} (VC_{5} C_{7} [p,q])&=&[((3+4)^{2} )^{(24pq-6p)} \times ((3+4)^{2} )^{(24pq-6p)} ]\\ &&\times [((2+3)^{2} )^{(12p)} \times ((3+3)^{2} )^{(12p) } ]\\ &=&5^{24p} \times 6^{24p} \times 7^{24p(4q-1)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HBII_{2} (VC_{5} C_{7} [p,q])&=&[((3\times 4)^{2} )^{(24pq-6p)} \times ((3\times 4)^{2} )^{(24pq-6p)} ]\\ &&\times [((2\times 3)^{2} )^{(12p)} \times ((3\times 3)^{2} )^{(12p)} ]\\ &=&3^{24p} \times 6^{24p} \times 12^{24p(4q-1)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(VC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HII_{b} (VC_{5} C_{7} [p,q])&=&[(\frac{2}{3+4} )^{(24pq-6p)} \times (\frac{2}{3+4} )^{(24pq-6p)} ]\times [(\frac{2}{2+3} )^{(12p)} \times (\frac{2}{3+3} )^{(12p)} ]\\ &=&(\frac{2}{7} )^{12p(4q-1)} \times (\frac{2}{5} )^{12p} \times (\frac{1}{3} )^{12p} . \end{eqnarray*}

2.7. Banhatti indices of \(HC_5C_7[p,q]\)

The molecular graphs of carbon nanotubes \(HC_5C_7[p, q]\) is shown in Figure 8.

Theorem 13. Let \(G\) be the \(HC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(B_{1} (HC_{5} C_{7} [p,q])=168pq+40p.\)
2. \(B_{2} (HC_{5} C_{7} [p,q])=288pq+32p.\)
3. \(HB_{1} (HC_{5} C_{7} [p,q])=1176pq+128p.\)
4. \(HB_{2} (HC_{5} C_{7} [p,q])=3456pq-184p.\)
5. \(H_{b} (HC_{5} C_{7} [p,q])=\frac{48}{7} pq+\frac{219}{35} p.\)

Proof. Let \(G=HC_{5} C_{7} [p,q].\) Then the edge set of \(HC_{5} C_{7} [p,q]\) can be partitioned as follows:
\(E_{4} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=2\},\)
\(E_{5} =\{ uv\in E(G):d_{G} (u)=2,d_{G} (v)=3\},\)
\(E_{6} =\{ uv\in E(G):d_{G} (u)=d_{G} (v)=3\},\)
such that \(|E_{4} |=p,\) \(|E_{5} |=8p\) and \(|E_{6} |=12pq-4p.\)
The edge degree partition is given in Table 7.

Table 7. Edge degree partition of \(HC_{5} C_{7} [p,q]\).

\(d_{G} (u),d_{G} (v):e=uv\in E(G)\)	\((2,2)\)	\((3,3)\)	\((2,3)\)
\(d_{G} (e)\)	\(2\)	\(4\)	\(3\)
Number of edges	\(p\)	\(12pq-4p\)	\(8p\)

Now

1. First K Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} B_{1} (HC_{5} C_{7} [p,q])&=&(p)[(2+2)+(2+2)]+(12pq-4p)[(3+4)+(3+4)]\\ &&+(8p)[(2+3)+(3+3)]\\ &=&168pq+40p. \end{eqnarray*}
2. Second K Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} B_{2} (HC_{5} C_{7} [p,q])&=&(p)[(2\times 2)+(2\times 2)]+(12pq-4p)[(3\times 4)+(3\times 4)]\\ &&+(8p)[(2\times 3)+(3\times 3)]\\ &=&288pq+32p. \end{eqnarray*}
3. First K hyper-Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{1} (HC_{5} C_{7} [p,q])&=&(p)[(2+2)^{2} +(2+2)^{2} ]+(12pq-4p)[(3+4)^{2} +(3+4)^{2} ]\\ &&+(8p)[(2+3)^{2} +(3+3)^{2} ]\\ &=&1176pq+128p. \end{eqnarray*}
4. Second K hyper-Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HB_{2} (HC_{5} C_{7} [p,q])&=&(p)[(2\times 2)^{2} +(2\times 2)^{2} ]+(12pq-4p)[(3\times 4)^{2} +(3\times 4)^{2} ]\\ &&+(8p)[(2\times 3)^{2} +(3\times 3)^{2} ]\\ &=&3456pq-184p. \end{eqnarray*}
5. K harmonic Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} H_{b} (HC_{5} C_{7} [p,q])&=&(p)[(\frac{2}{2+2} )+(\frac{2}{2+2} )]+(12pq-4p)[(\frac{2}{3+4} )+(\frac{2}{3+4} )]\\ &&+(8p)[(\frac{2}{2+3} )+(\frac{2}{3+3} )]\\ &=&\frac{48}{7} pq+\frac{219}{35} p. \end{eqnarray*}

Theorem 14. Let \(G\) be the \(HC_{5} C_{7} [p,q]\) nanotube. Then we have

1. \(BII_{1} (HC_{5} C_{7} [p,q])=2^{4p} \times 5^{8p} \times 6^{8p} \times 7^{8p(3q-1)}. \)
2. \(BII_{2} (HC_{5} C_{7} [p,q])=2^{4p} \times 3^{16p} \times 6^{8p} \times 12^{8p(3q-1)}. \)
3. \(HBII_{1} (HC_{5} C_{7} [p,q])=2^{8p} \times 5^{16p} \times 6^{16p} \times 7^{16p(3q-1)}. \)
4. \(HBII_{2} (HC_{5} C_{7} [p,q])=2^{16p} \times 5^{32p} \times 6^{32p} \times 7^{32p(3q-1)}. \)
5. \(HII_{b} (HC_{5} C_{7} [p,q])=(\frac{1}{2} )^{2p} \times (\frac{1}{3} )^{8p} \times (\frac{2}{5} )^{8p} \times (\frac{2}{7} )^{8p(3q-1)} .\)

Proof.

1. First multiplicative K Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} BII_{1} (HC_{5} C_{7} [p,q])&=&[(2+2)^{(p)} \times (2+2)^{(p)} ]\times [(3+4)^{(12pq-4p)} \times (3+4)^{(12pq-4p)} ]\\ &&\times [(2+3)^{(8p)} \times (3+3)^{(8p)} ]\\ &=&2^{4p} \times 5^{8p} \times 6^{8p} \times 7^{8p(3q-1)} . \end{eqnarray*}
2. Second multiplicative K Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} BII_{2} (HC_{5} C_{7} [p,q])&=&[(2\times 2)^{(p)} \times (2\times 2)^{(p)} ]\times [(3\times 4)^{(12pq-4p)} \times (3\times 4)^{(12pq-4p)} ]\\ &&\times [(2\times 3)^{(8p)} \times (3\times 3)^{(8p)} ]\\ &=&2^{4p} \times 3^{16p} \times 6^{8p} \times 12^{8p(3q-1)} . \end{eqnarray*}
3. First multiplicative K hyper-Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HBII_{1} (HC_{5} C_{7} [p,q])&=&[((2+2)^{2} )^{(p)} \times ((2+2)^{2} )^{(p)} ]\times [((3+4)^{2} )^{(12pq-4p)} \times ((3+4)^{2} )^{(12pq-4p)} ]\\ &&\times [((2+3)^{2} )^{(8p)} \times ((3+3)^{2} )^{(8p)} ]\\ &=&2^{8p} \times 5^{16p} \times 6^{16p} \times 7^{16p(3q-1)} . \end{eqnarray*}
4. Second multiplicative K hyper-Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HBII_{2} (HC_{5} C_{7} [p,q])&=&[((2\times 2)^{2} )^{(p)} \times ((2\times 2)^{2} )^{(p)} ]\times [((3\times 4)^{2} )^{(12pq-4p)} \times ((3\times 4)^{2} )^{(12pq-4p)} ]\\ &&\times [((2\times 3)^{2} )^{(8p)} \times ((3\times 3)^{2} )^{(8p)} ]\\ &=&2^{8p} \times 3^{32p} \times 6^{16p} \times 12^{16p(3q-1)} . \end{eqnarray*}
5. Multiplicative K harmonic Banhatti index of \(HC_{5} C_{7} [p,q]\) is \begin{eqnarray*} HII_{b} (HC_{5} C_{7} [p,q])&=&[(\frac{2}{2+2} )^{(p)} \times (\frac{2}{2+2} )^{(p)} ]\times [(\frac{2}{3+4} )^{(12pq-4p)} \times (\frac{2}{3+4} )^{(12pq-4p)} ]\\ &&\times [(\frac{2}{2+3} )^{(8p)} \times (\frac{2}{3+3} )^{(8p)} ]\\ &=&(\frac{1}{2} )^{2p} \times (\frac{1}{3} )^{8p} \times (\frac{2}{5} )^{8p} \times (\frac{2}{7} )^{8p(3q-1)} . \end{eqnarray*}

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

1. Introduction

Theory of characteristic modes (TCM) was developed by Garbacez [1] and gained importance after it was revisited in [2] by diagonalizing the impedance matrix of the body. Mathematically, we have [3, 4, 5, 6]

where \(Z=R+jX\), \(I_n\) and \(1+j\lambda_n\) represents the impedance, current eigen vector and eigen value respectively. Further simplification of the (2) gives It can also be represented as

Here \(J_n\) represents the current density eigen vector. From the Theorem of Reciprocity, if \(Z\) is linear symmetric operator then the Hermitian parts of \(Z\) (\(R\) and \(X\)) will also be linear symmetric operator [3]. Thus we concludes that the eigen values will be real and eigen vectors will be real and equiphasal. Eigen value gives us an information about the behavior of a modes at a particular frequency whether it will resonate or store electrical/mechanical energy. Modal significance (MS) is a parameter depending on the eigen value and gives us information about the contribution of a particular mode at a particular frequency. Mathematically it is given by

At resonance, the value of MS is equal to 1 and at 3 dB, the value of BW corresponds to 0.707. The radiation pattern associated with these real eigen modes are orthogonal to one another.

TCM is widely used for the analysis of various different types of antennas such as wire antennas [7], handset antennas [8], small antennas [9], dielectric antennas [10], printed MIMO antennas [11, 12, 13, 14, 15, 16] and slot antennas [17, 18, 19]

Multiple-input-multiple-output (MIMO) antennas are widely used for the enhancement of antenna capacity and this technology is used for 4G and 5G communications [20, 21]. The performance of such antennas can be degraded, if the antennas are not properly isolated [21, 22]. Recently TCM was used to enhance the isolation between the MIMO antenna elements.

The main contribution of this paper is to review the empirical method procedure (EMP) used for the isolation enhancement. We also review the isolation enhancement from the perspective of TCM. The short comings and the significance of the methods proposed in the literature with respect to TCM is highlighted. The breakup of this paper is: in section 2 we briefly discuss the different methods given in the literature from EMP method. In section 3 we discuss the isolation enhancement from the perspective of TCM. The conclusion is provided in section 4.

2. Isolation Enhancement using EMP

From EMP perspective the isolation between MIMO antennas can be enhanced by the use of meta surfaces or electromagnetic bandgap structures [23, 24, 25], parasitic elements [26, 27], neutralization line technique [28, 29], optimization of the antenna system configuration [30, 31], decoupling networks [32] and the use of Defected ground structures (DGS)[11, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Among all the aforementioned port isolation enhancement methods, DGS is the least complex and expensive.

Meta-material (single layer or double layer) are used for miniaturization and isolation. They suppress surface waves between the antennas resulting in an enhanced isolation [23, 24, 25]. Meta-materials normally involve larger and bulky formations that needs to be properly optimized. The isolation enhancement cant be obtained easily, if miniaturization is the major objective. Isolation improvement in [32] by the use of decoupling methods normally compromises of two stages: matching network and decoupling network. In decoupling network reactive elements are designed between the antenna feeding ports to improve the isolation, such networks require more space and generates additional cost. In addition, neutralization line technique was proposed in [28, 29]. These lines deliver certain amount of signal to the neighboring antenna that counteracts for the coupling between them. The major drawback is huge amount of optimization involved in the adjustment of the width, length and the location of connection point of these lines. More than one neutralization line is needed to improve the isolation over wide frequency bandwidth and this complicates the process further. Parasitic elements also uses the idea of field cancellation to enhance the isolation between the antennas [26, 27]. The major draw back is the optimization of the shape and the dimensions of parasitic element that play significant role in isolation enhancement.

In [33], a periodic S shaped DGS was used to improve the isolation between two patch antennas to \(23 \enspace dB\). Two DGS (T-shaped and line slot) were used to improve the isolation to \(18 \enspace dB\) over a wide frequency range of 3.1-10.6 GHz in [34]. Two DGS were used to address two different frequency bands. A DGS etched in a square ring fashion was used to improve isolation by \(7 \enspace dB\), for a square patch surrounded by a square ring patch design in [35]. An isolation was improved to \(17 \enspace dB\) for a \(2\) element \(4\) shaped dual band design, where the length of the rectangular slot and spirals in the DGS were used to tune the frequency band in [36]. In [37], an isolation of \(15 \enspace dB\) was achieved for a \(2\) element MIMO antenna by combining two isolation enhancement mechanisms that are orthogonal placement of antennas and introducing a slit in the ground. Isolation enhancement for a MIMO antenna system for more than \(2\) elements is difficult. Isolation was enhanced to \(12 dB\) for a very closely packed \(4\)-element MIMO design, where they used four wideband DGS with multi objective fractional factorial design [38]. A slitted pattern was etched between \(2\)-element PIFA and wire monopole design that enhanced the isolation to reach \(20 \enspace dB\) [39]. Moreover, the study was extended to \(4\)-element PIFA (aligned along a line) to get an isolation of \(12 \enspace dB\). Two DGS were used in [40] to improve isolation of a \(4\)-element MIMO design. A rectangular slot and a stair case slot were used to enhance the isolation to \(12 \enspace dB\) between the horizontal and verticals antennas respectively. For an \(8\)-element MIMO design, a very complicated DGS consisting of closed loop frequency selective surfaces and quad strips connected with a circular arc were used to improve isolation to \(15 \enspace dB\) respectively [42].

The main problem in the DGS method is to enhance isolation is the shape, size, number of DGS, position and the huge amount of optimization involved in the placement of DGS. We observed that from the perspective of EMP there is no systematic method to enhance the isolation. All of the available literature rely on the past experience and different parametric studies to obtain the enhanced isolation. This was the reason that the TCM was used for the isolation enhancement to develop a proper methodology for isolation enhancement.

3. Isolation Enhancement using TCM

The presence of any antenna or deformation in the chassis greatly affect the chassis modes [43, 44, 45, 46, 47]. TCM was used in [48, 49, 50, 51, 52] to enhance the isolation between the MIMO antenna designs. It was demonstrated in [48] that the current in PIFA is more localized as compared Monopole and this is the reason that PIFA antennas are narrow band as compared to Monopole antennas. At the same time MIMO PIFA antenna design has better isolation because of its localized current nature. For frequencies less than 1 GHz, the chassis starts contributing to the antenna performance because now the electrical length of the antenna is huge, whereas for MIMO antennas, the situation will worsen because now all the antenna will excite the chassis modes. For frequency of 1 GHz, the antenna having only one chassis mode have electric field maxima at the edge and electric field minima at the center. An electrical antenna placed at the edge will effectively excite the chassis modes while antenna placed at the center will not. Antennas placed at the mentioned position achieved a 5 dB more isolation as compared to the antennas placed at the edges of the chassis [48]. A magnetic and electrical antenna placed at the edges will have improved isolation because the electrical antenna will excite chassis currents while the magnetic antenna will not [49]. Co-located antennas were introduced in [49] to have very compact MIMO antenna design. To have better isolation, the co-located Magnetic and electric antenna excite the chassis currents in opposite direction. In [48, 49], only the behavior of the first mode using TCM was observed and analysis were based on it.

The selective excitation of characteristic modes that have orthogonal behavior can enhance the isolation. In [53], a feed network (consisting of 4 hybrid 180 couplers) and 4 capacitive coupler were designed to excite the four different modes of the antenna. The modes excited due to its orthogonal nature results in an enhanced isolation. The asymmetry in the ground plane produces a natural tilt and increase in directivity [54]. Two antennas were placed at asymmetrical ground plane such that both of them excite different characteristic modes, it will help in getting highly isolated and highly uncorrelated beams.

In [55], the lower order modes were separated by the use of decoupling network and GA was used to synthesize low quality factor MIMO antenna and quality factor is inversely proportional to the antenna bandwidth. A monopole (excited via CCE) and chassis (excited via ICE) combination was used to implement highly isolated MIMO design [56]. From the combination of CM, a new set of radiation modes can be formed, then the radiation modes are highly orthogonal to one another and thus resulting in highly uncorrelated beams [57].

In [58], out-of-band interference was improved by the use of TCM. For communication in aeroplane, antenna at 2.8 to 24 MHz is used with very strong power. The harmonics of this antenna are high power and thus affects the communication of other antennas. Such type of interference is called out-of-band interference. The methodology adopted was to calculate the modal mutual admittance (MMA) and modal self admittance of the antenna. The CMA of the higher frequency antenna (with low power) is calculated. Functional and non-functional modes of the first antenna is calculated where modes contributing to real communication are known as functional while the modes contributing to interference are known as non-functional and non-functional modes are blocked by inductor loading with the help of TCM. The value of the inductor shall be properly optimized to block the non-functional mode.

In [59], designs made with the application of TCM were compared with the empirically made designs. 5 designs were opted from different papers with 2 designs opted from TCM and 3 opted by empirical method. All of the selected designs were MIMO and they were compared in 7 different real time scenarios. The scenarios were: first of all, the designs performance was observed in the presence of a box, where 3 scenarios in which the antenna was hold by one hand and the 3 scenarios in which the antenna was hold by two hands. TCM design recorded 3 times high multiplexing efficiency as compared to conventional design and for all the 7 scenarios, TCM designs performed with ME of 1.6 dB better.

Now all these methods focus on the isolation enhancement for frequentness around 1 GHz [48, 49]. For a normal chassis, only one chassis mode of 1 GHz is present. The problem will escalate, if we consider frequency greater than 1.5 GHz as now more than one mode will be present. As the point where one mode was having current minima (a possible location for the placement of the second antenna), another mode current maxima lies over there. Thus we are left with no possible location.

In [12], a possible solution to this problem was proposed, where the DGS was used to enhance the isolation between two MIMO antennas. A methodology was also proposed to predict whether the isolation can be enhanced or not. The block diagram of the methodology is shown in the Figure 1. After the designing of MIMO antennas the characteristic modes were identified. The modes were classified as coupling and non-coupling modes based on the current distribution and its resemblance with the chassis current distribution in the presence of the excitation sources. Non-coupling modes are the one contributing to the radiation while the coupling modes are contributing to the port coupling thus degrading the isolation performance. If there exists a certain location on the chassis, where the DGS placed can block the coupling mode but at the same time does not effect the non-coupling mode the isolation between the antenna elements can be enhanced but if there does not exist any such location, the isolation between the antenna elements cannot be enhanced. The method was applied to different antenna designs and it achieved the most significant enhancement in the isolation as compared to other designs in the literature.

4. Conclusion

In this paper, a summary of different available techniques to enhance the isolation between MIMO antenna elements is presented. The shortcomings of the different EMP based approaches and the TCM based approaches to enhance the port isolation between the MIMO antenna elements are discussed. A new method to enhance the isolation between the MIMO antennas with the help of TCM is proposed. The method is able to predict whether the isolation can be enhanced or not.

Competing Interests

The author declare no conflict of interest.

The \(BiI_{3}\) is an inorganic compound which is the result of the reaction of iodine and bismuth, which inspired the enthusiasm for subjective inorganic investigations [1]. \(BiI_{3}\) is an excellent inorganic compound and is very useful in "qualitative inorganic analysis" [1, 2].

It was proved that $Bi$-doped glass optical strands are one of the most promising dynamic laser media. Different kinds of Bi-doped fiber strands have been created and have been used to construct Bi-doped fiber lasers and optical loudspeakers [3].

Layered \(BiI_{3}\) gemstones are considered to be a three-layered stack structure in which a plane of bismuth atoms is sandwiched between iodide particle planes to form a continuous \(I-Bi-I\) plane [4].

The periodic superposition of the diamond-shaped three layers forms \(BiI_{3}\) crystals with \(R\)-3 symmetry [5, 6]. A progressive stack of \(I-Bi-I\) layers forms a symmetric hexagonal structure [7] and jewel of \(BiI_{3}\) was integrated in [8].

In the unit cell (Figure 1), Main cycles are \(C_{4}^{1} ,C_{4}^{2}\) central cycles are \(C_{4}^{3} ,C_{4}^{6}\) and Base cycles are \(C_{4}^{4} ,C_{4}^{5}\)

Mathematical chemistry is an area of research in chemistry in which mathematical tools are used to solve problems of chemistry. Chemical graph theory is an important area of research in mathematically chemistry which deals with topology of molecular structure such as the mathematical study of isomerism and the development of topological descriptors or indices. Infect, TIs are real numbers attached with graph networks and graph of chemical compounds and has applications in quantitative structure-property relationships. TIs remain invariant upto graph isomorphism and help to predict many properties of chemical compounds, networks and nanomaterials, for example, viscosity, boiling points, radius of gyrations, etc without going to lab [9, 10, 11, 12].

Other emerging field is Cheminformatics, in which we use QSAR and QSPR relationship to guess biological activity and chemical properties of nanomaterial and networks. In these investigations, some Physico-chemical properties and TIs are utilized to guess the behavior of chemical networks [13, 14, 15, 16, 17]. Like TIs, polynomials also fund considerable applications in network theory and chemistry, for example, Hosoya polynomial, which is also known as Wiener polynomial, introduced in [18] plays an important role in computation of distance-based TIs. M-polynomial [19] was defined in 2015 and plays a similar role in computation of numerous degree-based TIs [20, 21, 22, 23, 24]. The M-polynomial contains precious information about degree-based TIs and many TIs can be computed from this simple algebraic polynomial. The first TI was defined in 1947 by Weiner during studying boiling point of alkanes [25]. This index is now known as Weiner index. Thus Weiner established the framework of TIs and the Wiener index is initially the first and most concentrated TI [26, 27].

The other oldest TI is Randić index (RI), given by Milan Randić [28] in 1975. After the success of Randić index, in the year 1988, the generalized version of Randić index was introduced [29, 30]. This version attracts both the mathematicians and chemists [31]. Numerous numerical properties of this simple TI are studied in [32]. For comprehensive study about this index, the book [33] can be of great help.

The RI is a most mainstream regularly connected and most concentrated among all other TIs. Numerous research papers and text books, for example, [34, 35, 36] are published in different academic journals on this TI. Two surveys on RI was written by Milan Randi\'{c} [37, 38] and three more surveys are written on this TI by different scientists [39, 40 41]. The reason behind the success of such a simple TI is as yet a puzzle, although some conceivable clarifications were given.

After Randi\'{c} index, the most studied TIs are 1st Zagreb index (ZI) and 2nd ZI [42, 43, 44, 45, 46]. The modified 2nd ZI was defined in [47]. Another TI is symmetric division (SDI) [48], Harmonic index (HI) [49, 50], augmented ZI [51].

In this article, we compute general form of several degree-based topological indices for Bismuth Tri-Iodide chains and Bismuth Tri-Iodide sheets. For example we compute first and second multiplicative Zagreb indices, multiplicative atomic bond connectivity index, sum connectivity index, modify Randi\'c index, etc.

2. Basic definitions and Literature Review

In mathematical chemistry, precisely speaking, in chemical-graph-theory (CGT), a molecular graph and graph network is a simple and connected graph, in which atoms represents vertices and chemical bonds represents edges. We reserve \(G\) for simple connected graph, \(E\) for edge set and \(V\) for vertex set throughout the thesis. The degree of a vertex \(u\) of graph \(G\) is the number of vertices that are attached with \(u\) and is denoted by \(d_{v}\). With the help of TIs, many properties of molecular structure can be obtained without going to lab [52]. The reality is, many research paper has been written on computation of degree-based indices and polynomials of different molecular structure and networks but only few work has been done so far on distance based indices and polynomials. In this paper, we aim to compute multiplicative degree-based TIs. Some indices related to Wiener's work are the first and second multiplicative Zagreb indices [53], respectively \[II_{1} \left(G\right)=\prod_{u\in V\left(G\right)}\left(d_{u} \right) ^{2} \] \[II_{2} \left(G\right)=\prod_{uv\in E\left(G\right)}d_{u} \cdot d_{v} \] and the Narumi-Katayama index [52] \[NK\left(G\right)=\prod_{u\in V\left(G\right)}d_{u} \] Like the Wiener index, these types of indices are the focus of considerable research in computational chemistry [54, 55, 56]. For example, in the year 2011, Gutman in [54] characterized the multiplicative Zagreb indices for trees and determined the unique trees that obtained maximum and minimum values for \(M_{1}(G)\) and \(M_{2}(G)\), respectively. Wang et al. in [57] extended the results of Gutman to the following index for k-trees, \[W_{1}^{s} \left(G\right)=\prod_{u\in V\left(G\right)}\left(d_{u} \right)^{s} .\] Notice that \(s = 1, 2\) is the Narumi-Katayama and Zagreb index, respectively. Based on the successful consideration of multiplicative Zagreb indices, Eliasi et al. [58] continued to define a new multiplicative version of the first Zagreb index as \[II_{1}^{*} \left(G\right)=\prod_{uv\in E\left(G\right)}\left(d_{u} +d_{v} \right) .\] Furthering the concept of indexing with the edge set, the first author introduced the first and second hyper-Zagreb indices of a graph [59]. They are defined as \[HII_{1} \left(G\right)=\prod_{uv\in E\left(G\right)}\left(d_{u} +d_{v} \right) ^{2} ,\] \[HII_{2} \left(G\right)=\prod_{uv\in E\left(G\right)}\left(d_{u} \cdot d_{v} \right)^{2} .\] In [60] Kulli et al. defined the first and second generalized Zagreb indices \[MZ_{1}^{a} \left(G\right)=\prod_{uv\in E\left(G\right)}\left(d_{u} +d_{v} \right) ^{\alpha} ,\] \[MZ_{2}^{a} \left(G\right)=\prod_{uv\in E\left(G\right)}\left(d_{u} \cdot d_{v} \right)^{\alpha} . \] Multiplicative sum connectivity and multiplicative product connectivity indices [61] are define as: \[SCII\left(G\right)=\prod_{uv\in E\left(G\right)}\frac{1}{\sqrt{d_{u} +d_{v} } } , \] \[PCII\left(G\right)=\prod_{uv\in E\left(G\right)}\frac{1}{\sqrt{d_{u} \cdot d_{v} } }. \] Multiplicative atomic bond connectivity index and multiplicative Geometric arithmetic index are defined as \[ABCII\left(G\right)=\prod_{uv\in E\left(\; G\right)}\sqrt{\frac{d_{u} +d_{v} -2}{d_{u} \cdot d_{v} } }, \] \[GAII\left(G\right)=\prod _{uv\in E\left(G\right)}\frac{2\sqrt{d_{u} \cdot d_{v} } }{d_{u} +d_{v} } ,\] \[GA^{a} II\left(G\right)=\prod _{uv\in E\left(G\right)}\left(\frac{2\sqrt{d_{u} \cdot d_{v} } }{d_{u} +d_{v} } \right) ^{\alpha}. \] Shigehalli and Kanabur [62] introduced following new degree-based topological indices:\\ Arithmetic-Geometric (AG1) index \(AG_{1} (G)=\sum_{uv\in E(G)}\frac{d_{u} +d_{v} }{2\sqrt{d_{u} d_{v} } } ,\) \(SK(G)=\sum_{uv\in E(G)}\frac{d_{u} +d_{v} }{2} ,\) \(AG_{1} (G)=\sum_{uv\in E(G)}\frac{d_{u} +d_{v} }{2\sqrt{d_{u} d_{v} } } ,\) \(SK_{2} (G)=\sum_{uv\in E(G)}\left(\frac{d_{u} +d_{v} }{2} \right)^{2}.\)

3. Computational Results

This section contains the main results. In this section we give formulae of multiplicative versions of degree-based TIs of Bismuth Tri-Iodide chains and Bismuth Tri-Iodide sheets. We also give formulae for some new degree-based TIs of Bismuth Tri-Iodide chains and Bismuth Tri-Iodide sheets.

3.1. Bismuth Tri-Iodide Chain

Theorem 1. Let \(G\) be the molecular graph of \(m-BiI_{3}\). Then

1. \(MZ_{1}^{\alpha} \left(G\right)=\left(7\right)^{\alpha (4p+8)} \times \left(8\right)^{\alpha (20p+4)} .\)
2. \(MZ_{2}^{\alpha} \left(G\right)=\left(2\right)^{\alpha (44p+16)} \times \left(3\right)^{\alpha (24p+12)} .\)
3. \(GA^{\alpha }II\left(G\right)=\left(\frac{2\sqrt{6} }{7} \right)^{\alpha (4p+8)} \times \left(\frac{\sqrt{3} }{2} \right)^{\alpha (20p+4)} .\)

Proof. Let \(G\) be the molecular graph of \(p-BiI_{3}\) bismuth tri-iodide chain. The edge set of \(p-BiI_{3}\) has following two partitions [1],
\(E_{1} =E_{\left\{1,6\right\}} =\left\{e=uv\in E\left(G\right) |d_{u} =1,\; d_{v} =6\right\},\)
\(E_{\left\{2,6\right\}} =\left\{e=uv\in E\left(G\right) |d_{u} =2,\; d_{v} =6\right\},\)
Such that
\(\left|E_{1} \left(G\right)\right|=4p+8,\)
\(\left|E_{2} \left(G\right)\right|=20p+4.\)
Now by definitions, we have

1. \begin{eqnarray*} MZ_{1}^{\alpha } \left(G\right)&=&\prod _{uv\in E\left(G\right)}\left(d_{u} +d_{v} \right) ^{\alpha } \\ &=&\left(1+6\right)^{\alpha (4p+8)} \times \left(2+6\right)^{\alpha (20p+4)} \\ & =&\left(7\right)^{\alpha (4p+8)} \times \left(8\right)^{\alpha (20p+4)} . \end{eqnarray*} 2. \begin{eqnarray*} MZ_{2}^{\alpha } \left(G\right)&=&\prod _{uv\in E\left(G\right)}(d_{u} \cdot d_{v} )^{\alpha }\\ &=&\left(1\times 6\right)^{\alpha \left(4p+8\right)} \times \left(2\times 6\right)^{\alpha (20p+4) }\\ &=&\left(2\times 3\right)^{\alpha (4p+8)} \times \left(4\times 3\right)^{\alpha (20p+4)} \\ &=&\left(2\right)^{\alpha (44p+16)} \times \left(3\right)^{\alpha (24p+12)}. \end{eqnarray*} 3. \begin{eqnarray*} GA^{\alpha } II\left(G\right)&=&\prod _{uv\in E\left(G\right)}\left(\frac{2\sqrt{d_{u} \cdot d_{v} } }{d_{u} +d_{v} } \right)^{\alpha } \\ &=&\left(\frac{2\sqrt{1\cdot 6} }{1+6} \right)^{\alpha (4p+8)} \times \left(\frac{2\sqrt{2\cdot 6} }{2+6} \right)^{\alpha (20p+8)} \\ &=& \left(\frac{2\sqrt{6} }{7} \right)^{\alpha (4p+8)} \times \left(\frac{\sqrt{3} }{2} \right)^{\alpha (20p+4)}. \end{eqnarray*}

Corollary 2. Let \(G\) be the molecular graph of \(p-BiI_{3}\). Then

1. \(MZ_{1} (G)=II_{2} \left(G\right){ =\; =}\left(2\right)^{44p+16} \times \left(3\right)^{24p+12} .\)
2. \(MZ_{2} (G)=II_{1}^{*} \left(G\right)=\left(7\right)^{4p+8} \times \left(8\right)^{20p+4} .\)
3. \(GAII\left(G\right){ \; =}\left(\frac{2\sqrt{6} }{7} \right)^{4p+8} \times \left(\frac{\sqrt{3} }{2} \right)^{20p+4} .\)

Proof. These results can be obtained immediately proved by taking \(\alpha =1\) in Theorem 1.

Corollary 3. Let \(G\) be the molecular graph of \(p-BiI_{3}\). Then

1. \(HII_{1} \left(G\right){ =}\left(7\right)^{2(4p+8)} \times \left(8\right)^{2(20p+4)} .\)
2. \(HII_{2} \left(G\right){ =}\left(2\right)^{88p+32} \times \left(3\right)^{48p+24} .\)

Proof. These results can be obtained immediately proved by taking \(\alpha =2\) in Theorem 1.

Corollary 4. Let \(G\) be the molecular graph of \(p-BiI_{3}\). Then

1. \(SCII\left(G\right)=\left(\frac{1}{\sqrt{7} } \right)^{4p+8} \times \left(\frac{1}{\sqrt{8} } \right)^{20p+4} .\)
2. \(PCII\left(G\right)=\left(\frac{1}{2} \right)^{22p+8} \times \left(\frac{1}{\sqrt{3} } \right)^{24p+12} .\)

Proof. These results can be obtained immediately proved by taking \(\alpha =-\frac{1}{2}\) in Theorem 1.

Theorem 5. Let \(G\) be the molecular graph of \(p-BiI_{3}\). Then \[ABCII\left(G\right)=\left(\sqrt{\frac{5}{6} } \right)^{4p+8} \times \left(\sqrt{\frac{1}{2} } \right)^{20p+4} .\]

Proof. Using the edge partition given in Theorem 1 and definition of multiplicative Atomic bond Connectivity index, we have \begin{eqnarray*}ABCII\left(G\right)&=&\prod _{uv\in E\left(G\right)}\sqrt{\frac{d_{u} +d_{v} -2}{d_{u} \cdot d_{v} } } \\ &=&\left(\sqrt{\frac{1+6-2}{1\cdot 6} } \right)^{4p+8} \times \left(\sqrt{\frac{2+6-2}{2\cdot 6} } \right)^{20p+4} \\\ &=&\left(\sqrt{\frac{5}{6} } \right)^{4p+8} \times \left(\sqrt{\frac{1}{2} } \right)^{20p+4} . \end{eqnarray*}

Theorem 6. Let \(G\) be the graph of \(p-BiI_{3}.\) Then

1. \(SC\left(G\right){ =}\left(\frac{4}{\sqrt{7} } +\frac{10}{\sqrt{2} } \right)p+\left(\frac{8}{\sqrt{7} } +\sqrt{2} \right).\)
2. \(AG_{1} \left(G\right){ =}\left(\frac{14}{\sqrt{6} } +\frac{40}{\sqrt{3} } \right)p+\left(\frac{28}{\sqrt{6} } +\frac{8}{\sqrt{3} } \right).\)
3. \(SK\left(G\right)=94p+44.\)
4. \(SK_{1} \left(G\right)=132p+48.\)
5. \(SK_{2} \left(G\right)=369p+162.\)
6. \(R'\left(G\right)=4p+2.\)

Proof. Using edge partition given in Theorem 1 and definitions, we have 1. \begin{eqnarray*} SC\left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{1}{\sqrt{d_{u} +d_{v} } } \\ &=&\frac{1}{\sqrt{7} } \left(4p+8\right)+\frac{1}{2\sqrt{2} } \left(20p+4\right) \\ &=&\left(\frac{4}{\sqrt{7} } +\frac{10}{\sqrt{2} } \right)p+\left(\frac{8}{\sqrt{7} } +\sqrt{2} \right). \end{eqnarray*} 2. \begin{eqnarray*} AG_{1} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} +d_{v} }{2\sqrt{d_{u} \times d_{v} } } \\ &=&\frac{7}{2\sqrt{6} } \left(4p+8\right)+\left(20p+4\right)\frac{2}{\sqrt{3} } \\ &=&\left(\frac{14}{\sqrt{6} } +\frac{40}{\sqrt{3} } \right)p+\left(\frac{28}{\sqrt{6} } +\frac{8}{\sqrt{3} } \right). \end{eqnarray*} 3. \begin{eqnarray*} SK\left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} +d_{v} }{2} \\ &=&\frac{7}{2} \left(4p+8\right)+4\left(20p+4\right) \\ &=&94p+44. \end{eqnarray*} 4. \begin{eqnarray*} SK_{1} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} \times d_{v} }{2} \\ &=&3\left(4p+8\right)+6\left(20p+4\right)\\ &=&132p+48.\end{eqnarray*} 5. \begin{eqnarray*}SK_{2} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\left(\frac{d_{u} +d_{v} }{2} \right) ^{2} \\ &=&\frac{49}{4} \left(4p+8\right)+16\left(20p+4\right) \\ &=&369p+162.\end{eqnarray*} 6. \begin{eqnarray*}R'\left(G\right)&=&\sum _{uv\in E\left(G\right)}\left(\frac{1}{\max \left\{d_{u} ,d_{v} \right\}} \right) \\ &=&\frac{1}{6} \left(4p+8\right)+\frac{1}{6} \left(20p+4\right) \\ &=&4p+2.\end{eqnarray*}

3.2. Bismuth Tri-Iodide sheet

In this section we compute several indices for Bismuth Tri-Iodide sheet.

Theorem 7. Let \(G\) be the molecular graph of \(BiI_{3} \left(p\times q\right)\). Then

1. \(MZ_{1}^{a} \left(G\right){ =}\left(7\right)^{\alpha (4p+4q+4)} \times \left(8\right)^{\alpha (12pq+8p+8q-4)} \times \left(9\right)^{\alpha (6pq-6q)} .\)
2. \(MZ_{2}^{a} \left(G\right)=\left(2\right)^{\alpha (30pq+20p+14q-4)} \times \left(3\right)^{\alpha (24pq+12p)} .\)
3. \({ GA}^{\alpha}{II}\left(G\right){ =}\left(\frac{2\sqrt{6} }{7} \right)^{\alpha (4p+4q+4)} \times \left(\frac{\sqrt{3} }{2} \right)^{\alpha (12pq+8p+8q-4)} \times \left(\frac{6\sqrt{2} }{9} \right)^{\alpha (6pq-6q)} .\)

Proof. Let \(G\) be the graph of \(BiI_{3} \left(p\times q\right)\) bismuth tri-iodide sheet. The edge set of \(BiI_{3} \left(p\times q\right)\) has following three partitions [1],
\(E_{1} =E_{\left\{1,6\right\}} =\left\{e=uv\in E\left(G\right){ |}d_{u} =1,\; d_{v} =6\right\},\)
\(E_{2} =E_{\left\{2,6\right\}} =\left\{e=uv\in E\left(G\right){ |}d_{u} =2,\; d_{v} =6\right\},\)
\(E_{3} =E_{\left\{3,6\right\}} =\left\{e=uv\in E\left(G\right){ |}d_{u} =3,\; d_{v} =6\right\},\)
such that
\(\left|E_{1} \left(G\right)\right|=4p+4q+4,\)
\(\left|E_{2} \left(G\right)\right|=12pq+8p+8q-4,\)
\(\left|E_{3} \left(G\right)\right|=6pq-6q.\)
Now by definition

1. \begin{eqnarray*}{ MZ}_{1}^{\alpha } \left(G\right)&=&\prod_{uv\in E\left(G\right)}\left(d_{u} +d_{v} \right)^{\alpha } \\ &=&\left(1+6\right)^{\alpha (4p+4q+4)} \times \left(2+6\right)^{\alpha (12pq+8p+8q-4)} \times \left(3+6\right)^{\alpha (6pq-6q)} \\ & =&\left(7\right)^{\alpha (4p+4q+4)} \times \left(8\right)^{\alpha (12pq+8p+8q-4)} \times \left(9\right)^{\alpha (6pq-6q)} . \end{eqnarray*} 2. \begin{eqnarray*}{ MZ}_{2}^{\alpha } \left(G\right)&=&\prod _{uv\in E\left(G\right)}(d_{u} \cdot d_{v} )^{\alpha } \\ &=&\left(6\right)^{\alpha (4p+4q+4)} \times \left(2\right)^{\alpha (12pq+8p+8q-4)} \times \left(3\right)^{\alpha (6pq-6q)} \\ &=&\left(2\right)^{\alpha (30pq+20p+14q-4)} \times \left(3\right)^{\alpha (24pq+12p)} .\end{eqnarray*} 3. \begin{eqnarray*} GA^{\alpha } II\left(G\right)&=&\prod _{uv\in E\left(G\right)}\left(\frac{2\sqrt{d_{u} \cdot d_{v} } }{d_{u} +d_{v} } \right)^{\alpha } \\ &=&\left(\frac{2\sqrt{1\cdot 6} }{1+6} \right)^{\alpha (4p+4q+4)} \times \left(\frac{2\sqrt{2\cdot 6} }{2+6} \right)^{\alpha (12pq+8p+8q-4)} \times \left(\frac{2\sqrt{3\cdot 6} }{3+6} \right)^{\alpha (6pq-6q)} \\ &=&\left(\frac{2\sqrt{6} }{7} \right)^{\alpha (4p+4q+4)} \times \left(\frac{\sqrt{3} }{2} \right)^{\alpha (12pq+8p+8q-4)} \times \left(\frac{6\sqrt{2} }{9} \right)^{\alpha (6pq-6q)} .\end{eqnarray*}

Corollary 8. Let \(G\) be the molecular graph of \(BiI_{3} \left(p\times q\right)\). Then

1. \(MZ_{1} (G)=II_{2} \left(G\right)=\left(2\right)^{30pq+20p+14q-4} \times \left(3\right)^{24pq+12p} .\)
2. \(MZ_{2} (G)=II_{1}^{*} \left(G\right){ =}\left(7\right)^{4p+4q+4} \times \left(8\right)^{12pq+8p+8q-4} \times \left(9\right)^{6pq-6q} .\)
3. \(GAII\left(G\right){ \; =}\left(\frac{2\sqrt{6} }{7} \right)^{4p+4q+4} \times \left(\frac{\sqrt{3} }{2} \right)^{12pq+8p+8q-4} \times \left(\frac{6\sqrt{2} }{9} \right)^{6pq-6q} .\)

Proof. These result can be obtained immediately proved by taking \(\alpha =1\) in Theorem 7.

Corollary 9. Let \(G\) be the molecular graph of \(BiI_{3} \left(p\times q\right)\). Then

1. \(HII_{1} \left(G\right){ =}\left(7\right)^{2(4p+4q+4)} \times \left(8\right)^{2(12pq+8p+8q-4)} \times \left(9\right)^{2(6pq-6q)} .\)
2. \(HII_{2} \left(G\right){ =}\left(2\right)^{4(15pq+10p+7q-2)} \times \left(3\right)^{24(2pq+p)} .\)

Proof. These result can be obtained immediately proved by taking \(\alpha =2\) in Theorem 7.

Corollary 10. Let \(G\) be the molecular graph of \(BiI_{3} \left(p\times q\right)\). Then

1. \(SCII\left(G\right)=\left(\frac{1}{\sqrt{7} } \right)^{4p+4q+4} \times \left(\frac{1}{\sqrt{8} } \right)^{12pq+8p+8q-4} \times \left(\frac{1}{\sqrt{9} } \right)^{6pq-6q} .\)
2. \(PCII\left(G\right){ =}\left(\frac{1}{\sqrt{2} } \right)^{30pq+20p+14q-4} \times \left(\frac{1}{\sqrt{3} } \right)^{24pq+12p} .\)

Proof. These result can be obtained immediately proved by taking \(\alpha =-\frac{1}{2}\) in Theorem 7.

Theroem 11. Let \(G\) be the molecular graph of \(BiI_{3} \left(p\times q\right)\). Then \[ABCII\left(G\right)=\left(\sqrt{\frac{5}{6} } \right)^{4p+4q+4} \times \left(\sqrt{\frac{1}{2} } \right)^{12pq+8p+8q-4} \times \left(\sqrt{\frac{7}{18} } \right)^{6pq-6q} .\]

Proof. \begin{eqnarray*} ABCII\left(G\right)&=&\prod _{uv\in E\left(G\right)}\sqrt{\frac{d_{u} +d_{v} -2}{d_{u} \cdot d_{v} } } \\ &=&\left(\sqrt{\frac{1+6-2}{1\cdot 6} } \right)^{4p+4q+4} \times \left(\sqrt{\frac{2+6-2}{2\cdot 6} } \right)^{12pq+8p+8q-4} \times \left(\sqrt{\frac{3+6-2}{3\cdot 6} } \right)^{6pq-6q} \\ &=&\left(\sqrt{\frac{5}{6} } \right)^{4p+4q+4} \times \left(\sqrt{\frac{1}{2} } \right)^{12pq+8p+8q-4} \times \left(\sqrt{\frac{7}{18} } \right)^{6pq-6q} . \end{eqnarray*}

Theroem 12. Let \(G\) be the graph of \(BiI_{3} \left(p\times q\right)\). Then

1. \(SC\left(G\right){ =}\left(\frac{6}{\sqrt{2} } +2\right)pq+\left(\frac{4}{\sqrt{7} } +\frac{4}{\sqrt{2} } \right)p+\left(\frac{4}{\sqrt{7} } +\frac{4}{\sqrt{2} } -2\right)q+\left(\frac{4}{\sqrt{7} } -\frac{2}{\sqrt{2} } \right).\)
2. \(AG_{1} \left(G\right){ =}\left(\frac{24}{\sqrt{3} } +\frac{9}{\sqrt{2} } \right)pq+\left(\frac{14}{\sqrt{6} } +\frac{16}{\sqrt{3} } \right)p+\left(\frac{14}{\sqrt{6} } +\frac{16}{\sqrt{3} } -\frac{9}{\sqrt{2} } \right)q+\left(\frac{14}{\sqrt{6} } -\frac{8}{\sqrt{3} } \right)\).
3. \(SK\left(G\right)=75pq+46p+19q-2.\)
4. \(SK_{1} \left(G\right)=126pq+60p+6q-12.\)
5. \(SK_{2} \left(G\right)=\frac{627}{2} pq+177p-\frac{111}{2} q-15.\)
6. \(R'\left(G\right)=3pq+2p+q.\)

Proof. Using the edge partition given in Theorem 7, we have

1. \begin{eqnarray*}SC\left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{1}{\sqrt{d_{u} +d_{v} } } \\ &=&\frac{1}{\sqrt{7} } \left(4p+4q+4\right)+\frac{1}{2\sqrt{2} } \left(12pq+8p+8q-4\right)+\frac{1}{3} \left(6pq-6q\right) \\ &=&\left(\frac{6}{\sqrt{2} } +2\right)pq+\left(\frac{4}{\sqrt{7} } +\frac{4}{\sqrt{2} } \right)p+\left(\frac{4}{\sqrt{7} } +\frac{4}{\sqrt{2} } -2\right)q+\left(\frac{4}{\sqrt{7} } -\frac{2}{\sqrt{2} } \right).\end{eqnarray*} 2. \begin{eqnarray*}AG_{1} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} +d_{v} }{2\sqrt{d_{u} \times d_{v} } } \\ &=&\frac{7}{2\sqrt{6} } \left(4p+4q+4\right)+\left(12pq+8p+8q-4\right)\frac{2}{\sqrt{3} } +\left(6pq-6q\right)\frac{3}{2\sqrt{2} } \\ &=&\left(\frac{24}{\sqrt{3} } +\frac{9}{\sqrt{2} } \right)pq+\left(\frac{14}{\sqrt{6} } +\frac{16}{\sqrt{3} } \right)p+\left(\frac{14}{\sqrt{6} } +\frac{16}{\sqrt{3} } -\frac{9}{\sqrt{2} } \right)q+\left(\frac{14}{\sqrt{6} } -\frac{8}{\sqrt{3} } \right).\end{eqnarray*} 3. \begin{eqnarray*}SK\left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} +d_{v} }{2} \\ &=&\frac{7}{2} \left(4p+4q+4\right)+4\left(12pq+8p+8q-4\right)+\frac{9}{2} \left(6pq-6q\right) \\ &=&75pq+46p+19q-2.\end{eqnarray*} 4. \begin{eqnarray*}SK_{1} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\frac{d_{u} \times d_{v} }{2} \\ &=&3\left(4p+4q+4\right)+6\left(12pq+8p+8q-4\right)+9\left(6pq-6q\right) \\ &=&126pq+60p+6q-12.\end{eqnarray*} 5. \begin{eqnarray*}SK_{2} \left(G\right)&=&\sum _{uv\in E\left(G\right)}\left(\frac{d_{u} +d_{v} }{2} \right) ^{2} \\ &=&\frac{49}{4} \left(4p+4q+4\right)+16\left(12pq+8p+8q-4\right)+\frac{81}{4} \left(6pq-6q\right) \\ &=&\frac{627}{2} pq+177p-\frac{111}{2} q-15.\end{eqnarray*} 6. \begin{eqnarray*}R'\left(G\right)&=&\sum _{uv\in E\left(G\right)}\left(\frac{1}{\max \left\{d_{u} ,d_{v} \right\}} \right) \\ &=&\frac{1}{6} \left(4p+4q+4\right)+\frac{1}{6} \left(12pq+8p+8q-4\right)+\frac{1}{6} \left(6pq-6q\right) \\ &=&3pq+2p+q.\end{eqnarray*}

4. Conclusions

In the present article, we computed closed form of 17 degree-based TIs for Bismuth Tri-Iodide chain and sheet. TIs thus calculated for these Bismuth Tri-Iodides can help us to understand the physical features, chemical reactivity, and biological activities. In this perspective, a TIs can be viewed as a score work which maps each sub-atomic structure to a real number and is utilized as descriptors of the particle under testing. These outcomes can likewise have a crucial influence in the assurance of the importance of Bismuth Tri-Iodide. For instance, it has been proved that the first Zagreb index is straightforwardly related with all out \(\pi\)-electron energy. Additionally Randic index is helpful for deciding physio-chemical properties of alkanes as seen by scientific expert Melan Randic in 1975. He saw the relationship between's the Randic index and a few physico--chemical properties of alkanes like, "enthalpies of formation, boiling points, chromatographic retention times, vapor pressure and surface areas" [52].

Competing Interests

The authors declare that they have no competing interests.