The study of topological indices, based on distance in a graph, was effectively employed in 1947 in chemistry by Weiner [1]. He introduced a distance-based topological index called the "Wiener index" to correlate properties of alkenes and the structures of their molecular graphs. These indices play a vital role in computational and theoretical aspects of chemistry in predicting material properties [2, 3-7]. Several algebraic polynomials have useful applications in chemistry, [8, 9].
A graph \(G\) is an ordered pair \((V,E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. A path from a vertex \(v\) to a vertex \(w\) is a sequence of vertices and edges that starts from \(v\) and stops at \(w\). The number of edges in a path is called the length of that path. A graph is said to be connected if there is a path between any two of its vertices. The distance \(d(u,v)\) between two vertices \(u\), \(v\) of a connected graph \(G\) is the length of a shortest path between them. Graph theory is contributing a lion's share in many areas such as chemistry, physics, pharmacy, as well as in industry [10]. We will start with some preliminary facts.
The first and the second Zagreb indices are defined as \begin{equation*} M_{1}(G)=\sum\limits_{u\in V(G)}(d_{u}+d_{v}), \end{equation*} \begin{equation*} M_{2}(G)=\sum\limits_{uv\in E(G)}d_{u}\times d_{u}, \end{equation*} \noindent For details see [11]. Considering the Zagreb indices, Fath-Tabar ([12]) defined first and the second Zagreb polynomials as $$M_{1}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}+d_{v}},$$ and $$M_{2}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}.d_{v}}.$$ The properties of \(M_{1}(G,x)\) , \(M_{2}(G,x)\) polynomials for some chemical structures have been studied in the literature [13, 14].
After that, in [15], the authors defined the third Zagreb index $$M_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}-d_{v}),$$ and the polynomial $$M_{3}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}-d_{v}}.$$ In the year 2016, [16] following Zagreb type polynomials were defined $$M_{4}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}(d_{u}+d_{v})},$$ $$M_{5}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{v}(d_{u}+d_{v})},$$ $$M_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{ad_{u}+bd_{v}},$$ $$M'_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{(d_{u}+a)(d_{v}+b)}.$$ Ranjini et al. [17] redefines the Zagreb index, ie, the redefined first, second and third Zagreb indices of graph \(G\). These indicators appear as $$Re ZG_{1}(G)=\sum\limits_{uv\in E(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}},$$ $$Re ZG_{2}(G)=\sum\limits_{uv\in E(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}},$$ and $$Re ZG_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}+d_{v})(d_{u}.d_{v}).$$ In this paper we compute other Zagreb polynomials and Redefined Zagreb indices of Möbiusbius Ladder. The Möbius ladder \(M_{n}\) which is a cubic circulant graph with an even number of vertices, formed from an n-cycle by adding edges (called "rungs") connecting opposite pair of vertices in the cycle. It is so-named because (with the exception of \(M_{6}=K_{3,3}\) has exactly \(\frac{n}{2}\) \(4\)-cycles which link together by their shared edges to form a topological Möbius strip. Möbius ladders can also be viewed as a prism with one twisted edge. Two different views of Möbius ladders have been shown in Figure 1. Möbius ladders have many applications in chemistry, chemical stereography, electronics and computer science. For our convenience, we view the Möbius ladder \(M_{n}\) which is a cubic circulant graph with an even number of vertices, formed from an n-cycle by adding edges (called "rungs") connecting opposite pair of vertices in the cycle.

2. Computational Results

Let In this section, we present our computational results.

Theorem 2.1 Let \(M_{n}\) be the Möbius Ladder. Then

1. \(M_{3}(M_{n},x)=3n,\)
2. \(M_{4}(M_{n},x)=3nx^{18},\)
3. \(M_{5}(M_{n},x)=3nx^{18},\)
4. \(M_{a,b}(M_{n},x)=3nx^{3a+3b},\)
5. \(M'_{M_{n},b}(G,x)=3nx^{(3+a)(3+b)}.\)

Proof. Let \(M_{n}\) be the Möbius Ladder. It is clear that \(M_{n}\) has only one partition of vertex set i.e, $$V_{1}=\{v\in V(M_{n}): d_{v}=3\},$$ The edge set of \(M_{n}\) has following one partition, $$E_{1}=E_{3,3}=\{e=uv\in E(M_{n}): d_{u}=3, d_{v}=3\},$$ Now, $$\mid E_{1}(M_{n})\mid=3n,$$ (1) \begin{eqnarray*} M_{3}(G,x)&=&\sum\limits_{uv\in E(M_{n})}x^{(d_{u}-d_{v})}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}x^{(3-3)}\\ &=&\mid E_{1}(M_{n})\mid \\ &=& 3n. \end{eqnarray*} (2) \begin{eqnarray*} M_{4}(G,x)&=&\sum\limits_{uv\in E(M_{n})}x^{d_{u}(d_{u}+d_{v})}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}x^{3(3+3)}\\ &=&\mid E_{1}(M_{n})\mid x^{18}\\ &=& 3nx^{18}. \end{eqnarray*} (3) \begin{eqnarray*} M_{5}(G,x)&=&\sum\limits_{uv\in E(M_{n})}x^{d_{v}(d_{u}+d_{v})}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}x^{3(3+3)}\\ &=&\mid E_{1}(M_{n})\mid x^{18}\\ &=& 3nx^{18}. \end{eqnarray*} (4) \begin{eqnarray*} M_{a,b}(G,x)&=&\sum\limits_{uv\in E(M_{n})}x^{(ad_{u}+bd_{v})}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}x^{(3a+3b)}\\ &=&\mid E_{1}(M_{n})\mid x^{3a+3b}\\ &=& 3nx^{3a+3b}. \end{eqnarray*} (5) \begin{eqnarray*} M'_{a,b}(G,x)&=&\sum\limits_{uv\in E(M_{n})}x^{(d_{u}+a)(d_{v}+b)}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}x^{(3+a)(3+b)}\\ &=&\mid E_{1}(M_{n})\mid x^{(3+a)(3+b)}\\ &=& 3nx^{(3+a)(3+b)}. \end{eqnarray*}

Theorem 2.2 Let \(M_{n}\) be the Möbius Ladder. Then,

1. \(Re ZG_{1}(M_{n})=2n\),
2. \(Re ZG_{2}(M_{n})= \frac{9}{2}n\),
3. \(Re ZG_{3}(M_{n})=162n\).

Proof.

(1) \begin{eqnarray*} Re ZG_{1}(M_{n})&=&\sum\limits_{uv\in E(M_{n})}\frac{d_{u}+d_{v}}{d_{u}d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}\frac{d_{u}+d_{v}}{d_{u}d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}\frac{3+3}{3.3}\\ &=&\mid E_{1}(M_{n})\mid \frac{6}{9}\\ &=&(3n)\frac{6}{9}\\ &=& 2np. \end{eqnarray*} (2) \begin{eqnarray*} Re ZG_{2}(G)&=&\sum\limits_{uv\in E(M_{n})}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}\frac{3.3}{3+3}\\ &=&(3n)\frac{3}{2}\\ &=& \frac{9}{2}n. \end{eqnarray*} (3) \begin{eqnarray*} Re ZG_{2}(G)&=&\sum\limits_{uv\in E(M_{n})}(d_{u}.d_{v})(d_{u}+d_{v})\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}(d_{u}.d_{v})(d_{u}+d_{v})\\ &=&\sum\limits_{uv\in E_{1}(M_{n})}(3.3)(3+3)\\ &=&\mid E_{1}(G)\mid 54\\ &=&3n(54)\\ &=& 162n. \end{eqnarray*}

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

Graph theory provides chemists with a variety of useful tools, such as topological indices. Molecular compounds are often modeled using molecular graphs. The molecular graph represents the structural formula of the compound in the form of graph theory, the vertices of which correspond to the atoms of the compound and the edges correspond to the chemical bonds [1].

Cheminformatics is a new area of research that integrates chemistry, mathematics, and information science. It studies the quantitative structure-activity (QSAR) and structure-property (QSPR) relationships [2, 3, 4, 5] used to predict the biological activity and properties of compounds. In the QSAR/QSPR study, the physical and chemical properties and topological indices such as Szeged index, Wiener index, Randić index, ABC index and Zagreb index etc were used to predict the biological activity of compounds. A molecular graph can be identified by topological index, polynomials, sequences or matrices [6].

The topological index is a number associated with the graph [7]. It represents the topological structure of the graph and is invariant under the automorphism of the graph. There are some major topological index categories, such as distance-based topological indices [8, 9], degree-based topological indices [10, 11], and counting-related polynomial and graph indices [7]. In these categories, the degree-based topological index is very important and plays a crucial role in chemical graph theory, especially in chemistry [12, 13, 14, 15]. More precisely, the topological index Top(G) of the graph is a number with the following characteristics: If a graph \(H\) is isomorphic to \(G\), then \(Top(H)=Top(G)\). The concept of topological index comes from Wiener [16], when he studied the boiling point of paraffin. He named this index as the path number. Later, the path number was renamed Wiener index.

Carbon nanotubes form an interesting class of non-carbon materials [17]. There are three types of nanotubes, namely chiral, zigzag and armchairs nanotubes [18]. These carbon nanotubes show significant mechanical properties [17]. Experimental studies have shown that they belong to the most rigid and elastic known materials [19]. Diudea [20] was the first chemist to consider the topology of nanostructures. \(HAC_{5}C_{6}C_{7}[p,q]\) [21]shown in Figure 1, is constructed by alternating \(C_{5}\), \(C_{6}\) and \(C_{7}\) carbon cycles. It is tube shaped material but we consider it in the form of sheet shown in Figure 2. The two dimensional lattice of \(HAC_{5}C_{6}C_{7}[p,q]\) consists of \(p\) rows and \(q\) periods. Here \(p\) denotes the number of pentagons in one row and \(q\) is the number of periods in whole lattice. A period consist of three rows (See references [22, 23]). Figures are taken from [24]. Figure 3 is 2D graph of \(HAC_{5}C_{6}C_{7}[p,q]\) and Figure 4 is line graph of \(HAC_{5}C_{6}C_{7}[p,q]\).

The aim of this paper is to compute Zagreb polynomials of the line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube. We also compute some degree-based topological indices of the line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube. A line graph has many useful applications in physical chemistry [25, 26] and is defined as: the line graph \(L(G)\) of a graph \(G\) is the graph each of whose vertex represents an edge of \(G\) and two of its vertices are adjacent if their corresponding edges are adjacent in \(G\).

2. Basic definitions and Literature Review

Throughout this article, we take \(G\) as a connected graph, \(V(G)\) is the vertex set and \(E(G)\) is the edge set. The degree of a vertex \(v\) is denoted by \(d_{v}\) and is eqult to the number of vertics attached to \(v\).
In the past two decades, a large number of topological indices have been defined and used for correlation analysis in theoretical chemistry, pharmacology, toxicology and environmental chemistry.
The first and second Zagreb indices are one of the oldest and most well-known topological indices defined by Gutman in 1972 and are given different names in the literature, such as the Zagreb group index, Sag. Loeb group parameters and the most common Zagreb index. The Zagreb index is one of the first indices introduced and has been used to study molecular complexity, chirality, ZE isomers and heterogeneous systems. The Zagreb index shows the potential applicability of deriving multiple linear regression models.
The first and the second Zagreb indices [27] are defined as \begin{equation*} M_{1}(G)=\prod\limits_{u\in V(G)}(d_{u}+d_{v}), \end{equation*} \begin{equation*} M_{2}(G)=\prod\limits_{uv\in E(G)}d_{u}\times d_{u}. \end{equation*} For details see [28]. Considering the Zagreb indices, Fath-Tabar ([29]) defined first and the second Zagreb polynomials as $$M_{1}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}+d_{v}},$$ and $$M_{2}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}.d_{v}}.$$ The properties of \(M_{1}(G,x)\) and \(M_{2}(G,x)\) for some chemical structures have been studied in the literature [30, 31].
After that, in [32], the authors defined the third Zagreb index $$M_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}-d_{v}),$$ and the polynomial $$M_{3}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}-d_{v}}.$$ In the year 2016, [33] following Zagreb type polynomials were defined $$M_{4}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}(d_{u}+d_{v})},$$ $$M_{5}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{v}(d_{u}+d_{v})},$$ $$M_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{ad_{u}+bd_{v}},$$ $$M'_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{(d_{u}+a)(d_{v}+b)}.$$ Ranjini et al. [34] redefined the Zagreb index, i.e, the redefined first, second and third Zagreb indices of graph \(G\). These indicators appear as $$Re ZG_{1}(G)=\sum\limits_{uv\in E(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}},$$ $$Re ZG_{2}(G)=\sum\limits_{uv\in E(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}},$$ and $$Re ZG_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}+d_{v})(d_{u}.d_{v}). $$ For details about topological indices and its applications we refer [35, 36, 37, 38, 39, 40, 41, 42, 43].

3. Main Results

In this section, we present our computational results.

Theorem 3.1. Let \(G\) be the line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube. Then

1. \(M_{3}(G,x)=2(38p-17)+2(6p+11)x,\)
2. \(M_{4}(G,x)=2x^{8}+12x^{12}+(6p+1)x^{18}+(12p+10)x^{21}+(70p-37)x^{32},\)
3. \(M_{5}(G,x)=2x^{8}+12x^{15}+(6p+1)x^{18}+(12p+10)x^{28}+(70p-37)x^{32},\)
4. \(M_{a,b}(G,x)=2x^{2(a+b)}+12x^{2a+3b}+(6p+1)x^{3(a+b)}+(12p+10)x^{3a+4b}\\ +(70p-37)x^{4(a+b)},\)
5. \(M'_{a,b}(G,x)=2x^{(a+2)(b+2)}+12x^{(a+2)(b+3)}+(6p+1)x^{(a+3)(b+3)}\\ +(12p+10)x^{(a+3)(b+4)}+(70p-37)x^{(a+4)(b+4)}.\)

Proof. Let \(G\) be the line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube where \(p\) denotes the number of pentagons in one row and \(q\) denotes the number of periods in whole lattice. The edge set of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) with \(p\geq 1\) and \(q=2\) has following five partitions, $$E_{1}=E_{2,2}=\{e=uv\in E(HAC_{5}C_{6}C_{7}[p,q]): d_{u}=2, d_{v}=2\},$$ $$E_{2}=E_{2,3}=\{e=uv\in E(HAC_{5}C_{6}C_{7}[p,q]): d_{u}=2, d_{v}=3\},$$ $$E_{3}=E_{3,3}=\{e=uv\in E(HAC_{5}C_{6}C_{7}[p,q]): d_{u}=3, d_{v}=3\},$$ $$E_{4}=E_{3,4}=\{e=uv\in E(HAC_{5}C_{6}C_{7}[p,q]): d_{u}=3, d_{v}=4\},$$ and $$E_{5}=E_{4,4}=\{e=uv\in E(HAC_{5}C_{6}C_{7}[p,q]): d_{u}=4, d_{v}=4\}.$$ Such that $$\mid E_{1}(G)\mid=2,$$ $$\mid E_{2}(G)\mid=12,$$ $$\mid E_{3}(G)\mid=6p+1,$$ $$\mid E_{4}(G)\mid=12p+10,$$ $$\mid E_{5}(G)\mid=70p+37.$$

(1) \begin{eqnarray*} M_{3}(G,x)&=& \sum\limits_{uv\in E(G)}x^{d_{u}-d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(G)}x^{2-2}+\sum\limits_{uv\in E_{2}(G)}x^{3-2}+\sum\limits_{uv\in E_{3}(G)}x^{3-3}+\sum\limits_{uv\in E_{4}(G)}x^{4-3}\\ &&\sum\limits_{uv\in E_{5}(G)}x^{4-4}\\ &=&\mid E_{1}(G)\mid +\mid E_{2}(G)\mid x+\mid E_{3}(G)\mid +\mid E_{4}(G)\mid x +\mid E_{5}(G)\mid\\ &=& 2+12x+(6p+1)+(12p+10)x+(70p-37)\\ &=& 2(38p-17)+2(6p+11)x. \end{eqnarray*} (2) \begin{eqnarray*} M_{4}(G,x)&=&\sum\limits_{uv\in E(G)}x^{d_{u}(d_{u}+d_{v})}\\ &=&\sum\limits_{uv\in E_{1}(G)}x^{2(2+2)}+\sum\limits_{uv\in E_{2}(G)}x^{2(2+3)}+\sum\limits_{uv\in E_{3}(G)}x^{3(3+3)}+\sum\limits_{uv\in E_{4}(G)}x^{3(3+4)}\\ &&\sum\limits_{uv\in E_{5}(G)}x^{4(4+4)}\\ &=&\mid E_{1}(G)\mid x^{8}+\mid E_{2}(G)\mid x^{10}+\mid E_{3}(G)\mid x^{18}+\mid E_{4}(G)\mid x^{28} +\mid E_{5}(G)\mid x^{32}\\ &=& 2x^{8}+12x^{12}+(6p+1)x^{18}+(12p+10)x^{21}+(70p-37)x^{32}. \end{eqnarray*} (3) \begin{eqnarray*} M_{5}(G,x)&=& \sum\limits_{uv\in E(G)}x^{d_{v}(d_{u}+d_{v})}\\ &=&\sum\limits_{uv\in E_{1}(G)}x^{2(2+2)}+\sum\limits_{uv\in E_{2}(G)}x^{3(3+2)}+\sum\limits_{uv\in E_{3}(G)}x^{3(3+3)}+\sum\limits_{uv\in E_{4}(G)}x^{4(4+3)}\\ &&\sum\limits_{uv\in E_{5}(G)}x^{4(4+4)}\\ &=&\mid E_{1}(G)\mid x^{8}+\mid E_{2}(G)\mid x^{15}+\mid E_{3}(G)\mid x^{18}+\mid E_{4}(G)\mid x^{28} +\mid E_{5}(G)\mid x^{32}\\ &=& 2x^{8}+12x^{15}+(6p+1)x^{18}+(12p+10)x^{28}+(70p-37)x^{32}. \end{eqnarray*} (4) \begin{eqnarray*} M_{a,b}(G,x)&=&\sum\limits_{uv\in E(G)}x^{ad_{u}+bd_{v}}\\ &=&\sum\limits_{uv\in E_{1}(G)}x^{2a+2b}+\sum\limits_{uv\in E_{2}(G)}x^{2a+3b}+\sum\limits_{uv\in E_{3}(G)}x^{3a+3b}+\sum\limits_{uv\in E_{4}(G)}x^{3a+4b}\\ &&\sum\limits_{uv\in E_{5}(G)}x^{4a+4b}\\ &=&\mid E_{1}(G)\mid x^{2(a+b)} +\mid E_{2}(G)\mid x^{2a+3b}+\mid E_{3}(G)\mid ^{3(a+b)} +\mid E_{4}(G)\mid x^{3a+4b} \\ &&+\mid E_{5}(G)\mid x^{4(a+b)}\\ &=& 2x^{2(a+b)}+12x^{2a+3b}+(6p+1)x^{3(a+b)}+(12p+10)x^{3a+4b}\\ &&+(70p-37)x^{4(a+b)}. \end{eqnarray*} (5) \begin{eqnarray*} M'_{a,b}(G,x)&=& \sum\limits_{uv\in E(G)}x^{(d_{u}+a)(d_{v}+b)}\\ &=&\sum\limits_{uv\in E_{1}(G)}x^{(2+a)+(2+b)}+\sum\limits_{uv\in E_{2}(G)}x^{(a+2)(3+b)}+\sum\limits_{uv\in E_{3}(G)}x^{(a+3)(3+b)}\\ &&+\sum\limits_{uv\in E_{4}(G)}x^{(3+a)(4+b)}+\sum\limits_{uv\in E_{5}(G)}x^{(4+a)+(4+b)}\\ &=&\mid E_{1}(G)\mid x^{(a+2)(b+2)} +\mid E_{2}(G)\mid x^{(a+2)(b+3)}+\mid E_{3}(G)\mid x^{(a+3)(b+3)}\\ && +\mid E_{4}(G)\mid x^{(a+3)(b+4)} +\mid E_{5}(G)\mid x^{(a+4)(b+4)}\\ &=& 2x^{(a+2)(b+2)}+12x^{(a+2)(b+3)}+(6p+1)x^{(a+3)(b+3)}\\ &&+(12p+10)x^{(a+3)(b+4)}+(70p-37)x^{(a+4)(b+4)}. \end{eqnarray*}

Theorem 3.2. For every \(p\geq 1\) and \(q=2\) consider \(G\) be the the graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube. Then

1. \(Re ZG_{1}(G)=46p,\)
2. \(Re ZG_{1}(G)= \frac{1187}{7}p+\frac{2727}{70}\),
3. \(Re ZG_{1}(G)=2(5146p-1725)\).

Proof. From the edge partition of line graph of \(HAC_{5}C_{6}C_{7}[p,q]\) nanotube given in Theorem 3.1

(1)\begin{eqnarray*} Re ZG_{1}(G)&=&\sum\limits_{uv\in E(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}+\sum\limits_{uv\in E_{2}(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}+\sum\limits_{uv\in E_{3}(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}\\ &&+\sum\limits_{uv\in E_{4}(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}+\sum\limits_{uv\in E_{5}(G)}\frac{d_{u}+d_{v}}{d_{u}d_{v}}\\ &=&\mid E_{1}(G)\mid +\mid E_{2}(G)\mid \frac{5}{6}+\mid E_{3}(G)\mid \frac{6}{9}+\mid E_{4}(G)\mid \frac{7}{12} +\mid E_{5}(G)\mid\frac{8}{16}\\ &=&2 +(12)\frac{5}{6}+(6p+1)\frac{6}{9}+(12p+10) \frac{7}{12} +(70p+37)\frac{8}{16}\\ &=& 46p. \end{eqnarray*} (2)\begin{eqnarray*} Re ZG_{2}(G)&=&\sum\limits_{uv\in E(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &=&\sum\limits_{uv\in E_{1}(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}+\sum\limits_{uv\in E_{2}(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}+\sum\limits_{uv\in E_{3}(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &&+\sum\limits_{uv\in E_{4}(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}+\sum\limits_{uv\in E_{5}(G)}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &=&\mid E_{1}(G)\mid +\mid E_{2}(G)\mid \frac{6}{5}+\mid E_{3}(G)\mid \frac{9}{6}+\mid E_{4}(G)\mid \frac{12}{7} +\mid E_{5}(G)\mid\frac{16}{8}\\ &=&2 +(12)\frac{6}{5}+(6p+1)\frac{9}{6}+(12p+10) \frac{12}{7} +(70p+37)\frac{16}{8}\\ &=& \frac{1187}{7}p+\frac{2727}{70}. \end{eqnarray*} (3) \begin{eqnarray*} Re ZG_{2}(G)&=&\sum\limits_{uv\in E(G)}(d_{u}.d_{v})(d_{u}+d_{v})\\ &=&\sum\limits_{uv\in E_{1}(G)}(d_{u}.d_{v})(d_{u}+d_{v})+\sum\limits_{uv\in E_{2}(G)}(d_{u}.d_{v})(d_{u}+d_{v})\\ &&+\sum\limits_{uv\in E_{3}(G)}(d_{u}.d_{v})(d_{u}+d_{v})+\sum\limits_{uv\in E_{4}(G)}(d_{u}.d_{v})(d_{u}+d_{v})\\ &&+\sum\limits_{uv\in E_{5}(G)}(d_{u}.d_{v})(d_{u}+d_{v})\\ &=&16 \mid E_{1}(G)\mid +30 \mid E_{2}(G)\mid+54 \mid E_{3}(G)\mid + 84 \mid E_{4}(G)\mid +128 \mid E_{5}(G)\mid\\ &=&32 +30(12)+54(6p+1)+84 (12p+10)+128(70p+37)\\ &=& 2(5146p-1725). \end{eqnarray*}

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

The nanotechnology is an area of intense scientific research in which nanomaterials are synthesized, characterized and applied. Nanoparticles are the clusters of atoms, ions or molecules who's size range is < 100nm. These include fullerenes, metal clusters, proteins etc. Many nanomaterial are formed by metals like Cobalt (Co), Copper (Cu), Zinc (Zn), Magnesium (Mg), Titanium (Ti), Gold (Au) and Silver (Ag). Metal oxides nanoparticles can be prepared easily.Due to their small size they differ from the bulk materials. The nanoparticles are being used for different purposes i.e. for medical treatments, in different branches of industry productions such as solar and oxide fuel batteries for energy storage, for incorporation into various materials of daily use e.g. cosmetics or clothes. The Zinc oxide (ZnO) nanoparticles has been widely used in the light emitting diodes, solar cells, piezoelectric transducers, gas sensors and as catalysts for long time. ZnO is one of the n-type semiconductor which have bandgap of about 3.37ev and has been used in different fields of catalysis, sensors, electronic devices, solar cells. As the ZnO nanoparticles have smaller size so they have large surface area.They have been synthesized by different methods so far. In 2010, they were synthesized from aqueous solutions in the form of equiaxed nanoparticles [1].They show some photocatalytic properties and are widely used in sunscreen cosmetics, clothes and also for the degradation of environmental pollutants. They have some biological activities also. In 2012, the antitumor activity of the photostimulated ZnO nanoparticles was studied. They were used with cisplatin and the results showed the increased antitumor activity [2]. In 2013, ZnO nanoparticles were also synthesized by using supercritical methanol in good yield [3]. They can be synthesized with combination of transition metals e.g. through the electrolysis process in which Zinc plate is used as anode in sodium tungstate solution [4].They are very effective against UV blocking and for in vivo toxicity of the polymer coated. The ZnO nanoparticles coated with chitosan (ZnO-CTS) and polyethylene glycol (PEG) which is a synthetic polymer (ZnO-PEG) have also been synthesized and their photocatalytic activity was studied which showed the increased photocatalytic activity and stability and which also showed the better ultraviolet absorption efficiency [5].They can also be produced through hydrothermal method in which microwave is used. In this process by changing in the power and the time for microwave irradiation cause different morphological effects on the ZnO nanoparticles [6].The ZnO nanoparticles were observed to cause eosinophilc airway inflammation in mice [7].

2. Method and materials

2.1. Materials

Zinc Chloride (ZnCl2), Sodium Hydroxide (NaOH). All these chemicals were of AR grade and purchased from Sigma Aldrich.

2.2. Procedure

We took 5g of Zinc Chloride (ZnCl2) and dissolved in small amount of distilled water to make its solution. This Solution was titrated with 0.5M Sodium Hydroxide (NaOH) solution unless white precipitate appeared. Water was added in this precipitate and then washed with distilled water 5 times. This precipitate was filtered. The precipitate was separated and then put it in hot air oven at a temperature of 1050C for 3 hours. Then it was grind with mortar and pestle for 15 minutes and then put it in muffle furnace at 300°C for 5 hours for calcination. After calcination its color changed to cream yellow from white. Then was grind again and stored in glass bottle for further process.

3. Biological synthesis of Zno nanoparticles

3.1. Materials

Zinc chloride (ZnCl2), Sodium hydroxide (NaOH), Deionized water, Aspargillus niger. All the chemicals were of AR grade and purchased from Sigma Aldrich.

3.2. Procedure

In this method first of all salt solution was prepared by dissolving 5g of salt in deionized water and mixed with 2 g crushed powder of Aspargillus niger and stirred for 30 minutes. Then the resultant material was placed in dark for 3 days. After 3 days the solution was filtered. The pale white filtrate was obtained. This filtrate was dried in hot air oven at105°C and then calcined in muffle furnace at 550°C for 3 hours .The material was grind and stored for further analysis.

4. Results and discussion

For the characterization of prepared Zinc Oxide nanoparticles, XRD is one of the best technique. It characterizes the purity and phase of the nanomaterial. XRD gives detail of diffraction angle, the interlayer spacing and mainly the crystallite size. The XRD used in our work was Bruker D8 advance.Figure.1 shows the XRD pattern of zinc oxide through chemical method. The peaks at \(2\theta=32.0\), 34.5, 36.0, 47.5, 56.0, 63.0, 67.0, 68.0 and 69.0 which correspond to (100), (002), (101), (102), (110), (103), (200), (112) and (201) crystalline planes of ZnO and were in good agreement with the JCPDS Card no. 01-079-2205. All these peaks exactly match with the literature [8, 9] which clearly indicates the formation of zinc oxide. The average crystallite size of ZnO NP's was 71.5 nm, calculated using Scherrer equation based on the full width at half-maximum of the (101) diffraction plane

The XRD pattern of biologically synthesized Zinc Oxide nanoparticles is shown in Figure 2. The peaks at 2? position of 31.76, 34.42, 36.25, 47.53, 56.60, 62.86, 66.37, 67.96 and 69.09 represent the ZnO nanoparticles of crystallite size of 40.8 nm size successfully prepared through biological method.

Then Zinc Oxide nanoparticleswere characterized by the Scanning Electron Microscopy (SEM) for their surface morphology. The SEM used in this experiment was FEI Quanta 450 FEG. Fig.3 shows FESEM images of Zinc Oxide nanoparticles at different magnifications. The particles are agglomerated. The particles are not enough separated which show the presence of weak physical forces. The results were also matched with the earlier studies [10, 11].

The Zinc Oxide nanoparticles synthesized through biological methods were characterized through SEM, TESCAN, VEGA3. The SEM images of biologically synthesized nanoparticles are shown in Figure 4. Which also show the spherical and agglomerated ZnO nanoparticles formation.

The Elemental analysis was carried out through EDX. Figure.5 shows the EDX spectra of Zinc Oxide nanoparticles prepared from chemical method.This shows the highest weight % of 57.03 for Zn which clearly shows the formation of ZnO nanoparticles.

The elemental analysis of biologically synthesized Zinc Oxide nanoparticles was performed on EDX Oxford. Figure.6 shows the EDX spectra of Zinc Oxide nanoparticles prepared from biological method. In this case the Zn weight % is 85.9 while oxygen 8.6%.Which is clear indication of formation of the Zinc Oxide nanoparticles.

5. Conclusion

In summary, The Zinc Oxide nanoparticles were successfully synthesized by two different methods i.e. chemical and biological. For biological synthesis Aspargillus niger was used. The product obtained was characterized through different analytical techniques like XRD, SEM and EDX.The obtained results weretallied with literature. It was revealed that the ZnO nanoparticles can be prepared from Aspargillus niger in sufficient quantity with high purity.

Acknowledgement

This work was impossible without the help and guidelines of my supervisor Dr. Zaheer Ahmad, Chemistry Department University of Wah, Wah Cantt and co-supervisor Dr. Tariq Mahmood, NCP, Islamabad. I cordially thank them for their support and guidance throughout this research work.

Competing Interests

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

1. Introduction

A number, polynomial or a matrix can uniquely identify a graph. A topological index is a numeric number associated to a graph which completely describes the topology of the graph, and this quantity is invariant under the isomorphism of graphs. The degree-based topological indices are derived from degrees of vertices in the graph. These indices have many correlations to chemical properties. In other words, a topological index remains invariant under graph isomorphism. The study of topological indices, based on distance in a graph, was effectively employed in 1947 in chemistry by Weiner [1]. He introduced a distance-based topological index called the "Wiener index" to correlate properties of alkenes and the structures of their molecular graphs. Recent progress in nano-technology is attracting attention to the topological indices of molecular graphs, such as nanotubes, nanocones, and fullerenes to cut short experimental labor. Since their introduction, more than 140 topological indices have been developed, and experiments reveal that these indices, in combination, determine the material properties such as melting point, boiling point, heat of formation, toxicity, toughness, and stability [2]. These indices play a vital role in computational and theoretical aspects of chemistry in predicting material properties [3, 4, 5, 6, 7, 8].

Several algebraic polynomials have useful applications in chemistry, such as the Hosoya Polynomial (also called the Wiener polynomial) [9]. It plays a vital role in determining distance-based topological indices. Among other algebraic polynomials, the M-polynomial introduced recently in 2015 [10] plays the same role in determining the closed form of many degree-based topological indices. Other famous polynomials are the first Zagreb polynomial and the second Zagreb polynomial.

A graph \(G\) is an ordered pair \((V, E)\), where \(V\) is the set of vertices and \(E\) is the set of edges. A path from a vertex \(v\) to a vertex \(w\) is a sequence of vertices and edges that starts from \(v\) and stops at \(w\). The number of edges in a path is called the length of that path. A graph is said to be connected if there is a path between any two of its vertices. The distance \(d(u, v)\) between two vertices \(u\) and \(v\) of a connected graph \(G\) is the length of a shortest path between them. Graph theory is contributing a lion's share in many areas such as chemistry, physics, pharmacy, as well as in industry [11]. We will start with some preliminary facts. The first and second multiplicative Zagreb indices [12], respectively \begin{equation} MZ_{1}(G)=\prod\limits_{u\in V(G)}(d_{u})^{2}, \end{equation} \begin{equation} MZ_{2}(G)=\prod\limits_{uv\in E(G)}d_{u}. d_{u}, \end{equation} and the Narumi-Kataymana index [13] \begin{equation} NK(G)=\prod\limits_{u\in V(G)}d_{u}, \end{equation} Like the Wiener index, these types of indices are the focus of considerable research in computational chemistry [13, 14, 15, 16, 17]. For example, in 2011, Gutman [14] 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. and the last author [17] then extended Gutman's result to the following index for k-trees, \begin{equation} W^{s}_{1}(G)=\prod\limits_{u\in V(G)}(d_{u})^{s}. \end{equation} 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. [18] continued to define a new multiplicative version of the first Zagreb index as \begin{equation} MZ^{\ast}_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u}), \end{equation} Furthering the concept of indexing with the edge set, the researhers introduced the first and second hyper-Zagreb indices of a graph [19] as \begin{equation} HII_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u})^{2}, \end{equation} \begin{equation} HII_{2}(G)=\prod\limits_{uv\in E(G)}(d_{u}.d_{u})^{2}, \end{equation} In [20]. Kulli et al. defined the first and second generalized Zagreb indices \begin{equation} MZ^{a}_{1}(G)=\prod\limits_{uv\in E(G)}(d_{u}+d_{u})^{a}, \end{equation} \begin{equation} MZ^{a}_{2}(G)=\prod\limits_{uv\in E(G)}(d_{u}.d_{u})^{a}, \end{equation} Multiplicative sum connectivity and multiplicative product connectivity indices [21] are define as: \begin{equation} SCII(G)=\prod\limits_{uv\in E(G)}\frac{1}{(d_{u}+d_{u})}, \end{equation} \begin{equation} PCII(G)(G)=\prod\limits_{uv\in E(G)}\frac{1}{(d_{u}.d_{u})}, \end{equation} Multiplicative atomic bond connectivity index and multiplicative Geometric arithmetic index are defined as \begin{equation} ABCII(G)=\prod\limits_{uv\in E(G)}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}, \end{equation} \begin{equation} GAII(G)=\prod\limits_{uv\in E(G)}\frac{2\sqrt{d_{u}.d_{u}}}{d_{u}+d_{u}}, \end{equation} \begin{equation} GA^{a}II(G)=\prod\limits_{uv\in E(G)}\left(\frac{2\sqrt{d_{u}.d_{u}}}{d_{u}+d_{u}}\right)^{a} \end{equation}

Definition 1.1. Let \(n\),\(m\), and \(a_{1}, a_{2},...,a_{m}\) be positive integers, where \(1\leq a_{i}\leq \left\lfloor\frac{n}{2}\right\rfloor\) and \(a_{i}\neq a_{j}\) for all \(1\leq i\leq j\leq m\) . An undirected graph with the set of vertices \(V=\{v_{1},v_{2},...,v_{n}\}\) and the set of edges \(E=\{v_{i}v_{i+a_{i}}: 1\leq i\leq n, 1\leq j\leq m\}\) where the indices being taken modulo \(n\), is called the circulant graph, and is denoted by \(C_{n}(a_{1}, a_{2},...,a_{m})\) The graph of \(C_{11}(1, 2, 3)\) is shown in Figure 1.

2. Computational Results

In this section, we present our computational results.

Theorem 2.1. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) be the Circulant graph. Then

1. \(MZ^{a}_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))=2(n-1)^{an}\),
2. \(MZ^{a}_{2}(C_{n}(a_{1}, a_{2},...,a_{m}))= (n-1)^{2an}\),
3. \(G^{a}AII(C_{n}(a_{1}, a_{2},...,a_{m}))=1\).

Proof. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) , where \(n=3,4,...,n\) and \(1\leq a_{i}\leq \left\lfloor\frac{n}{2}\right\rfloor\) and \(a_{i}\neq a_{j}\) , when \(n\) is odd to be circulant graph. It is clear form the graph of \(C_{n}(a_{1}, a_{2},...,a_{m})\) that the circulant gaph has only one partition of the vertex set i.e $$V_{1}=\{v\in V(C_{n}(a_{1}, a_{2},...,a_{m})): d_{v}=n\}.$$ The edge partition of \(C_{n}(a_{1}, a_{2},...,a_{m})\) has the following partition, $$E_{1}=E_{n-1,n-1}=\{e=uv\in E(C_{n}(a_{1}, a_{2},...,a_{m})): d_{u}=d_{v}=n-1\}.$$ Now, $$\mid E_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))\mid=n$$ 1. \begin{eqnarray*} MZ^{a}_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))&=& \prod\limits_{uv\in E(G)}(d_{u}+d_{v})^{a}\\ &=&(d_{u}+d_{v})^{a|E(C_{n}(a_{1}, a_{2},...,a_{m}))|}\\ &=& ((n-1)+(n-1))^{an}\\ &=& 2(n-1)^{an}. \end{eqnarray*} 2. \begin{eqnarray*} MZ^{a}_{2}(C_{n}(a_{1}, a_{2},...,a_{m}))&=& \prod\limits_{uv\in E(G)}(d_{u}\times d_{v})^{a}\\ &=&(d_{u}+d_{v})^{a|E(C_{n}(a_{1}, a_{2},...,a_{m}))|}\\ &=& ((n-1)\times (n-1))^{an}\\ &=& (n-1)^{2an}. \end{eqnarray*} 3. \begin{eqnarray*} G^{a}AII(C_{n}(a_{1}, a_{2},...,a_{m}))&=& \prod\limits_{uv\in E(C_{n}(a_{1}, a_{2},...,a_{m}))}\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a}\\ &=&\left(\frac{2\sqrt{d_{u}d_{v}}}{d_{u}+d_{v}}\right)^{a|E_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))|}\\ &=&\left(\frac{2\sqrt{(n-1)^{2}}}{2(n-1)}\right)^{an}\\ &=&1. \end{eqnarray*}

Corollary 2.2. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) be the Circulant graph. Then

1. \(MZ_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))=2(n-1)^{n}\),
2. \(MZ_{2}(C_{n}(a_{1}, a_{2},...,a_{m}))= (n-1)^{2n}\),
3. \(GAII(C_{n}(a_{1}, a_{2},...,a_{m}))=(1)^{n}=1\).

Proof. We get our result by putting \(\alpha=1\) in the Theorem 2.1.

Corollary 2.3. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) be the Circulant graph. Then

1. \(H II_{1}(C_{n}(a_{1}, a_{2},...,a_{m}))=2(n-1)^{2n}\),
2. \(H II_{2}(C_{n}(a_{1}, a_{2},...,a_{m}))= (n-1)^{4n}\).

Proof. We get our desired results by putting \(\alpha=2\) in Theorem 2.1.

Corollary 2.4. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) be the Circulant graph. Then

1. \(X II(C_{n}(a_{1}, a_{2},...,a_{m}))=\left(\frac{1}{\sqrt{2(n-1)}}\right)^{n} \),
2. \(\chi II(C_{n}(a_{1}, a_{2},...,a_{m}))=\left(\frac{1}{n-1}\right)^{n}\).

Proof. We get our desired results by putting \(\alpha=\frac{-1}{2}\) in Theorem 2.1.

Corollary 2.5. Let \(C_{n}(a_{1}, a_{2},...,a_{m})\) be the Circulant graph. Then $$ABCII(C_{n}(a_{1}, a_{2},...,a_{m}))= \left(\sqrt{\frac{2(n-2)}{(n-1)^{2}}}\right)^{n}. $$

Proof. By using the partition in Theorem 2.1, we have \begin{eqnarray*} ABCII(C_{n}(a_{1}, a_{2},...,a_{m}))&=&\prod\limits_{uv\in E(C_{n}(a_{1}, a_{2},...,a_{m}))}\sqrt{\frac{d_{u}+d_{u}-2}{d_{u}.d_{u}}}\\ &=&\left(\sqrt{\frac{(n-1)+(n-2)-2}{(n-1)^{2}}}\right)^{n} \\ &=& \left(\sqrt{\frac{2(n-2)}{(n-1)^{2}}}\right)^{n}. \end{eqnarray*}

Competing Interests

The authors declare that they have no competing interests.

1. Introduction

Pseudoephedrine is the most popular active nasal decongestant due to its effectiveness and relatively mild side effects [1]. In recent years, it has become increasingly difficult to obtain pseudoephedrine in many states because of its use as a precursor for the illegal drug N-methyl amphetamine (also known under various names including crystal meth, meth ice, etc.) [2]. Fexofenadine, 2-[4-(1-hydroxy-4-{4-[hydroxyl (biphenyl) methyl] piperidin-1-yl} butyl) phenyl]-2-methylpropanoic acid is a highly selective peripheral histamine H1 receptor antagonist used in the treatment of allergic diseases such as allergic rhinitis and chronic urticarial.

Fexofenadine is the active derivative of the antihistamine terfenadine, with no anti-cholinergic or alpha 1-adernergic receptor-blocking effects and without severe cardiac side effects of terfenadine [3, 4]. In literature survey many analytical methods have been reported for the estimating of individual Pseudoephedrine hydrochloride [5, 6],few HPLC assay and dissolution methods have been reported for determination of fexofenadine in pharmaceutical preparation [7].The estimation of fexofenadine in biological fluids using liquid chromatography with mass spectrometry [8], ionspray tandem mass spectrometry [9], electronspray tandem mass spectrometry [10], UV detection [11, 1], [Christopher MR et al., (1995)], but no work has so far been carried out for the simultaneous determination of Fexo. HCl and Pseudo.HCl in combine tablet dosage form by UV-vis spectrophotometry. The aim of the present study is to develop and validate a new and economical method for the simultaneous determination of Fexo. HCl and Pseudo.HCl in combine pharmaceutical tablet preparations. The method was validated in compliance with ICH guidelines [12].

2. Materials and method

2.1 Chemicals and reagents

The reference standards Fexofenadine HCl and Pseudoephedrine HCl (99.4%) pure were received as a gift sample from Java Pharmaceutical Kot Lakhpat, Lahore. Organic solvents ethanol and methanol (AR grade) was procured from Merck Chemical. Distilled water was used throughout the study. Sample tablets Fexet-D (Fexo. HCl and Pseudo. HCl 60mg/120mg) were purchased from local market Lahore., Pakistan.

2.2 Apparatus

TLC tank was used for the separation of both drugs. A single beam UV-Spectrophotometer (Cecil CE 2041, 2000 series) was used for the measurement. Analytical balance (JS-110, Japan) was used to weigh the sample and standard (Fexo. HCl and Pseudo.HCl ) material.

2.3 Selection of common solvent

Main criteria for media selection was solubility and stability, i.e. Fexo. HCl and Pseudo. HCl should be soluble as well as stable for sufficient time in selected media Pseudo. HCl show solubility in distilled water and ethanol (1:1) and Fexo. HCl in methanol respectively. It is economical and hence selected for analysis.

3. Methodology

3.1 Thin layer Chromatograph

The grind powder of tablet was dissolved in methanol and filtered the solution with fliter paper. The obtained concentrated solution was subjected on precoated silica gel plates. The polarity system CHCl3/n-Hexane (80:20%) was developed for TLC. The two A.P.I’s were separated which showed the Rf values 0.25±0.01 and 0.67+/-0.01 for Fexo. HCl and Pseudo. HCl respectively. As media selection, distilled water and ethanol (1:1) were used for Pseudo. HCl and methanol for Fexo.HCl, in which both the drugs were soluble and stable for a sufficient time.

3.2 Preparation of standard stock solutions and calibration curve

An accurately weighed 100mg each Fexo.HCl and Pseudo. HCl (reference standards) were transferred to two 100 ml volumetric flask separately and dissolved in methanol and distilled water/ethanol (1:1) individually and make up the volume up to the mark with the same solvent to obtain standard solution having concentration 1000ppm. Magnetic stirrer was used for better dissolution. Further 10ppm dissolution was made by taking 1ml from each of the above solution and make up the volume to 100ml with methanol and distilled water/ethanol (1:1). The working standard solutions 10 μg/mL of Fexo.HCl and Pseudo. HCl were scanned in the entire uv range 200-400nm to obtain the absorption spectra Fexo. HCl and Pseudo. HCl showed maximum absorption at 220nm and 247nm respectively. Further six dilutions from each stock solution were made with their respective solvents in the range 4-14 μg/mL and 5-30 μg/mL for Fexo. HCl and Pseudo.HCl respectively. The absorbance of resulting solutions were measured at respective \(\lambda_{max}\) and plotted a calibration curve against concentration to get the linearity and regression equation as shown in Fig.1.

3.3 Application of the Proposed Procedure for the Determination in Tablets

The proposed method was applied to determine the concentration of active drug in tablets dosage form. Twenty tablets were weighed and crushed to fine powder, drug equivalent to 60 mg and 120 mg Fexo. HCl and Pseudo. HCl was weighed and taken in 100 ml volumetric flask and make up the volume with methanol and distilled water/ethanol (1:1) respectively. The above solution was filtered by using Whattmann filter paper No. 41. From the above filtrate 10 ppm solution of each active drug was made and subjected for analysis. Analysis procedure was repeated six times with tablet formulation. Aliquot was scanned in the UV range (200-400nm).The amount of drug present in the tablets was calculated from the standard graphs as given in Table I.

3.4 Assay Measurement

The mean assay results of six sample tablets were comparable with claimed value. The obtained results are presented in Table-I and percentage was found to be 100.1% and 100.6% respectively.

Table I - Assay Determination of Fexo .HCl and Pseudo. HCl from its Tablet

Sample Tablet		Label Claimed	Amount Found mg ∕Tab.	Mean % Assay
Fexo .HCl	220 nm	60 mg	60.1 mg	100.1 %
Pseudo. HCl	247 nm	120 mg	120.8 mg	100.6 %

4 Method Validation

The developed method was validated by following parameters as provided by ICH.

4.1 Specificity

The sample and the standard spectra were scanned to check the specificity of the method. There was not found any interference of the excipients for the determination of Fexo. HCl and Pseudo. HCl which confirmed the method is highly specified for the estimation of Fexo. HCl and Pseudo.HCl in its tablet formulation.

4.2 Linearity

Various concentrations of the both analyte were made to measure the linearity of the method. The concentration range was 4-14 ppm at 220nm for Fexo. HCl and 5-30 ppm at 247nm for Pseudo.HCl. A calibration curve of absorbance versus concentration was plotted. Regression analysis was the confirmation of linearity of this method

Table II -Regression Analysis Fexo. HCl and Pseudo. HCl

Samples	Parameters	Results
FexofendineHCl	Regression equation	Y= 0.0643x + 0.9370
	Regression coefficient	\(R^2\) =0.9574
	Correlation coefficient	R = 0.9993
PseuoephdrineHCl	Regression equation	Y= 0.0843x + 0.0219
	Regression coefficient	\(R^2\) =0.9987
	Correlation coefficient	R = 0.9992

4.3 Accuracy

To ensure the accuracy of method, recovery study was performed by preparing six sample solutions of both drugs and added a known amount of active drug to each sample solution then measuring absorbance at 220nm and 247nm respectively. The % recovery was calculated along with SD and % RSD as listed in table II&III.

Table III - % Recovery Result of Fexofenadine HCl

Samples after addition	Absorbance after addition	% Recovery Assay
Sample 1	0.717	99.88 %
Sample 2	0.708	97.6 %
Sample 3	0.713	99.86 %
Sample 4	0.721	100.2%
Sample 5	0.715	99.23 %
Sample 6	0.720
Mean % Recovery Assay± SD		98.99 %± 0.943
Mean% Recovery Assay± %RSD		99.29 %± 0.95

Table IV - % Recovery result of Pseudoephedrine HCl

Samples after addition	Absorbance after addition	% Recovery Assay
Sample 1	0.178	99.88 %
Sample 2	0.172	97.6 %
Sample 3	0.179	99.86 %
Sample 4	0.176	100.2%
Sample 5	0.177	99.23 %
Sample 6	0.174	98.99 %
Mean % Recovery Assay± SD		99.29 %± 0.941
Mean% Recovery Assay± RSD		99.29 %± 0.93

4.4 Precision

Two different tablet solution was taken to measure the precision of method (Tablet-A and Tablet-B) and comparing the value of mean percentage assay with the proposed assay. The mean percentage (%) assay of the tablets was found to be very close to the proposed assay value (100.1 % and 100.6 %) respectively both for Fexo. HCl and Pseudo.HCl. Hence the assay method was found to be precise.

Table V - % Assay result of Tablet-A and Tablet –B

Fexofenadine HCl	Absorbance of Tablet-A	% Assay of Tablet-A	Absorbance of Tablet-B	% Assay of Tablet-B
Sample 1	0.717	99.6 %	0.716	99.26%
Sample 2	0.708	99.51 %	0.702	99.53%
Sample 3	0.713	100.2 %	0.709	98.77%
Sample 4	0.721	99.02%	0.714	100.11%
Sample 5	0.715	100.48 %	0.719	99.75%
Sample 6	0.720	98.65 %	0.723	100.8%
Mean % Recovery Assay± SD	-	99.57 %±0.689	-	99.53%± 0.851
Mean % Recovery Assay± %RSD	-	99.57 %±0.691	-	99.53%±0.854

Table VI - % Assay result of Tablet-A and Tablet –B

Fexofenadine HCl	Absorbance of Tablet-A	% Assay of Tablet-A	Absorbance of Tablet-B	% Assay of Tablet-B
Sample 1	0.178	99.6 %	0.180	99.26%
Sample 2	0.172	99.51 %	0.173	99.53%
Sample 3	0.179	100.2 %	0.182	98.77%
Sample 4	0.176	99.02%	0.184	100.11%
Sample 5	0.177	100.48 %	0.181	99.75%
Sample 6	0.174	98.65 %	0.183	100.8%
Mean % Recovery Assay± SD	-	99.57 %±0.689	-	99.53%±0.851
Mean % Recovery Assay± %RSD	-	99.54 %±0.687	-	99.51%± 0.751

4.5 Robustness

Robustness was measured by changing the wavelength as 220 + 1nm and 247 + 1nm (+ 1nm 221nm, 219 nm and 248 and 246 nm). The effect of change in wavelength was observed and mean percentage assay was calculated at two different wavelengths and was found to be very close to the proposed assay value (100.1% and 100.6 %), thus the robustness parameter was passed by the sample tablets.

Table VII - % Assay Result of \(\lambda_{max+ 1}\) and \(\lambda_{max–1}\) Conditions

Fexofenadine HCl	Wavelength plus condition \(\lambda_{max+ 1}\)		Wavelength subtract condition\(\lambda_{max- 1}\)
	Absorbance	% Assay	Absorbance	% Assay
Sample 1	0.715	100.4 %	0.719	99.75 %
Sample 2	0.720	98.65%	0.723	100.8 %
Sample 3	0.713	100.2%	0.716	99.26%
Sample 4	0.721	99.02%	0.702	99.53%
Sample 5	0.708	99.5 %	0.714	100.11
Sample 6	0.717	99.6%	0.709	98.75%
Mean % Recovery Assay± SD	-	99.34 %±0.836	-	99.43%±0.913
Mean % Recovery Assay± %RSD	-	99.57 %±0.841	-	99.53%± 0.915

Table VIII - % Assay result of \(\lambda_{max+ 1}\) and \(\lambda_{max1}\) Conditions

Fexofenadine HCl	Wavelength plus condition \(\lambda_{max+1}\)		Wavelength subtract condition \(\lambda_{max-1}\)
	Absorbance	% Assay	Absorbance	% Assay
Sample 1	0.178	99.60 %	0.176	99.02 %
Sample 2	0.172	99.51 %	0.177	100.48 %
Sample 3	0.179	100.2%	0.174	98.65%
Sample 4	0.180	99.26%	0.184	100.11 %
Sample 5	0.173	99.53%	0.181	99.75 %
Sample 6	0.182	98.77%	0.183	100.8 %
Mean Recovery Assay± SD	-	99.34%±0.836	-	99.48%±0.911
Mean %Recovery Assay± %RSD	-	99.34%±0.841	-	99.48%±0.914

4.6 Ruggedness

Ruggedness was determined by analysing the sample preparations on two different days to check the Ruggedness of the method. The mean percentage % assay at twoconsecutive days was found to be very close to the proposed value (100.1% &100.6 %) respectively.

Table IX - % Assay Result of Two Different Days Say Day -1 and Day -2

Samples	Day-1		Day-2
Fexofenadine HCl	Absorbance	% Assay	Absorbance	% Assay
Sample 1	0.715	100.4 %	0.721	99.02%
Sample 2	0.719	99.75 %	0.702	99.53%
Sample 3	0.723	100.8 %	0.717	99.6%
Sample 4	0.713	100.2%	0.714	100.11%
Sample 5	0.720	98.65%	0.709	98.75%
Sample 6	0.716	99.26%	0.708	99.5 %
Mean %Recovery Assay±SD	-	99.39%±0.763	-	99.58%± 0.968
Mean %Recovery Assay± %RSD	-	99.39 %±0.763	-	99.58%± 0.968

Table X - % Assay result of two different days say Day -1 and Day

Samples	Day-1		Day-2
Pseudoephedrine HCl	Absorbance	% Assay	Absorbance	% Assay
Sample 1	0.178	99.87 %	0.182	98.77 %
Sample 2	0.173	99.38 %	0.179	99.88 %
Sample 3	0.184	100.4%	0.183	100.03%
Sample 4	0.181	99.51%	0.176	99.75 %
Sample 5	0.177	99.02%	0.180	98.38 %
Sample 6	0.172	98.16%	0.174	101 %
Mean %Recovery Assay± SD	-	99.39%±0.763	-	99.78%±0.968
Mean %Recovery Assay± %RSD	-	99.39%±0.767	-	99.78%±0.971

4.7 LOD and LOQ

The LOD and LOQ of Fexo.HCl and Pseudo. HCl in its tablet formulation by proposed method were determined using calibration standards. LOD and LOQ were calculated and the results are shown in Table 11.

Table 11. Limit of Detection (LOD) and Limit of Quantitation (LOQ) Results

Samples	Parameters	Results
FexofendineHCl	Slope	0.0643
	Standard deviation	0.021
	LOD	1.077 ppm
	LOQ	3.265 ppm
PseuoephdrineHCl	Slope	0.0843
	Standard deviation	0.03
	LOD	1.174 ppm
	LOQ	3.585 ppm

5. Results and Discussions

Fexofenadine HCl and Pseudoephedrine HCl are used as antiallergic and nasal decongestant respectively. The present article deals with the development and validation of a new and an economical method for the simultaneous determination of Fexo. HCl and Pseudo. HCl in combine tablet dosage form by uv-vis spectrophotometry and TLC. The two drugs are present combine in the ratio of (1:2) which poses a problem in their assay determination. The main problem was to separate the two active ingredient from a single bilayered tablet because both the A.P.I’s were soluble in the same solvents. The published method was carried out on HPLC which is time taking and expensive for the routine analysis of pharmaceutical sectors. However , we have made an attempt to separate both the drugs by TLC and estimate it by UV-Visible spectrophotomtere which is more reliable and economical. The grind powder of tablet was dissolved in methanol and filtered the solution with fliter paper. The obtained concentrated solution was subjected on precoated silica gel plates. The polarity system CHCl3/n-Hexane (80:20%) was developed for TLC. Both the A.P.I’s were separated which showed the Rf values 0.25±0.01 and 0.67+/-0.01 for Fexo. HCl and Pseudo. HCl respectively. Visualization of single spot on TLC plate confirmed the purification of compounds. For media selection, distilled water and ethanol (1:1) were used for Pseud. HCl and methanol was used for Fexo. HCl in which both the drugs were soluble and stable for sufficient time. Both drugs were measured at 220nm and 247nm respectively, where they showed maximum absorbance. Beer Lambert’s law was obeyed at concentration range 4-14ppm and 5-30 ppm for Fexo. HCl and Pseudo.HCl respectively. A linearity curve was calibrated by concentration versus absorbance. Fexo.HCl (Y=0.0643x+0.9370) was measured with correlation coefficient r =0.9574 and Pseudo. HCl (Y=0.0843x+0.0219) with correlation coefficient r =0.9992. The results of analysis have been validated statistically and recovery studies was carried out as 99.19% and 99.29% which were close to the assay value 100.1% and 100.6% respectively. Precision of the method was measured which showed results for SD (99.57 %±) and % RSD (99.53 %±), The LOD (1.077ppm) and LOQ (3.265ppm) following ICH guidelines were measured which were found to be within limit. The proposed method was found to be specific, stable, linear, accurate, precise, and reproducible therefore it can be used for routine quality control analysis of these drugs in either alone or in combined pharmaceutical dosage forms.

Conclusion

The present method is specific, linear and reproducible thus it can be used for routine quality control analysis of these drugs in either alone or in combined pharmaceutical dosage forms.

Acknowledgement

The authors are gratified to Jawa Pharmaceutical, 112/10 Industrial area, Kot Lakhpat Lahore, Pakistan for providing a gift sample of Fexofenadine HCl and Pseudoephedrine HCl and facilities for the study. We are also thankful to Mr. Baqir Jawa (CEO) Jawa Pharmaceutical for his valuable cooperation during the whole research work.

Competing Interests

The authors declare that they have no competing interests.