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Open Journal of Mathematical Analysis

Vol. 2 (2018), Issue 1, pp. 66–73
ISSN: 2616-8111 (Online) 2616-8103 (Print)
DOI: 10.30538/psrp-oma2018.0012

Zagreb polynomials and redefined Zagreb indices for the line graph of carbon nanocones

Saba Noreen\(^1\), Atif Mahmood
Department of Mathematics and Statistics, The University of Lahore, Lahore Pakistan.; (S.N)
Department of Mathematics and Statistics, The University of Lahore, (Pakpattan Campus) Lahore Pakistan.; (A.M)
\(^{1}\)Corresponding Author; sabaalvi77@yahoo.com

Copyright © 2018 Saba Noreen, Atif Mahmood. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: April 9, 2018 – Accepted: May 30, 2018 – Published: June 30, 2018

Abstract

A line graph has many useful applications in physical chemistry. Topological indices are numerical parameters associated to a structure and, in combination, determine properties of the concerned material. In this paper, we compute the closed form of Zagreb polynomilas of all generalized class of carbon nanocones and compute important degree-based topological indices.

Keywords:

Zagreb-polynomial; Degree-based index; Carbon nanocone.

1. Introduction

Chemical graph theory is a subject which connects Mathematics, chemistry and graph theory. A topological index is a numeric number associated with molecular graph and this number correlate certain physico-chemical properties of chemical compounds. The topological indices such as the Wiener index, first and second Zagreb index, modified Zagreb index, Randi´c index and symmetric division index, Harmonic index, Invers sum index, Augmented Zagreb index, etc. are useful in prediction of bioactivity of the chemical compounds [1, 2, 3, 4, 5]. These indices capture the overall structure of compound and predict chemical properties such as strain energy, heat of formation, and boiling points etc. Carbon nanocones have been observed since 1968 or even earlier [6], on the surface of naturally occurring graphite. The importance of carbon nanostructures is due to their potential use in many applications including gas sensors, energy storage, nanoelectronic devices, biosensors and chemical probes [7]. Carbon allotropes such as carbon nanocones and carbon nanotubes have been proposed as possible molecular gas storage devices [8, 9]. More recently, carbon nanocones have gained increased scientific interest due to their unique properties and promising uses in many novel applications such as energy and hydrogen-storage [10]. Figure 1 and figure 2 are carbon nenocones.

Figure 1. Carbon Nenocone \(CNC_{k}[5]\).

Figure 2. The Molecular graph of \(CNC_{k}[5]\).

The molecular graph of \(CNC_{k}[n]\) nanocones have conical structures with a cycle of length \(k\) at its core and \(n\) layers of hexagons placed at the conical surface around its center as shown in following figure 3.

Figure 3. Carbon nenocone \(CNC_{k}[n]\).

The line graph \(L(G)\) of a graph \(G\) is the graph each of whose vertex represents an edge of \(G\) [11, 12, 13,14] and two of its vertices are adjacent if their corresponding edges are adjacent in \(G\).

Figure 2. The line graph ofCarbon nenocone \(CNC_{k}[n]\).

In the present report, we gave closed form of Zagreb-polynomials of Carbon nenocones. We also compute some degree-based topological indices. In [15], authors computed Hosoya polynomial and related distance based indices for \(CNC_{7}[n]\). In [16] authors computed the Vertex PI, Szeged and Omega Polynomials of Carbon Nanocones \(CN_{4}[n]\). Similarly many partial results about topological indices have been obtained about some particular classes of Nanocones. We however present most general results about complete families of nanocones. Our results present nice generalizations of many existing partial results. Let \(G\) be a connected graph. The vertex and edge sets are denoted by \(V(G)\) and \(E(G)\), respectively. For every vertex \(v\in V(G)\), degree of \(v\) is number of vertices attached with it. The first and the second Zagreb indices (cf. [17]) are defined as $$M_{1}(G)=\sum\limits_{uv\in E(G)}(d_{u}+d_{v})$$ and $$M_{2}(G)=\sum\limits_{uv\in E(G)}(d_{u}.d_{v}).$$ Considering the Zagreb indices, Fath-Tabar [18] defined first and the second Zagreb polynomials $$M_{1}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}+d_{v}}$$ and $$M_{2}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}.d_{v}}.$$ The properties of first and second Zagreb polynomials for some chemical structures have been studied in the literature [19]. After that, in [20], the authors defined the third Zagreb index $$M_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}-d_{v})$$ and Zagreb polynomials $$M_{3}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}-d_{v}}.$$ In the year 2016, [21] following Zagreb type polynomials were defined
  • \(M_{4}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{u}(d_{u}+d_{v})}\),
  • \(M_{5}(G,x)=\sum\limits_{uv\in E(G)}x^{d_{v}(d_{u}+d_{v})}\),
  • \(M_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{ad_{u}+bd_{v}}\),
  • \(M^{'}_{a,b}(G,x)=\sum\limits_{uv\in E(G)}x^{(a+d_{u})+(b+d_{v})}\).
Ranjini et al. [22] redefines 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\(ZG_{2}(G)=\sum\limits_{uv\in E(G)}\frac{d_{u}d_{v}}{d_{u}+d_{v}}\),
  • Re\(ZG_{3}(G)=\sum\limits_{uv\in E(G)}(d_{u}.d_{v})(d_{u}+d_{v})\).

2. Main Results

In this section, we will present our computational results

Theorem 2.1 Let \(L(CNC_{k}[n])\) be the line graph of Carbon Nanocones \(CNC_{k}[n]\) . Then

  1. \(M_{1}(L(CNC_{k}[n]),x)=2kx^{5}+k(2n-1)x^{6}+2knx^{7}+3kn^{2}x^{8}\)
  2. \(M_{2}(L(CNC_{k}[n]),x)=2kx^{6}+k(2n-1)x^{9}+2knx^{12}+3kn^{2}x^{16}\)
  3. \(M_{3}(L(CNC_{k}[n]),x)=k[3n^2+2n-1]+2k(n+1)x\)
  4. \(M_{4}(L(CNC_{k}[n]),x)=2kx^{10}+k(2n-1)x^{18}+2knx^{21}+3kn^{2}x^{32}\)
  5. \(M_{5}(L(CNC_{k}[n]),x)=2kx^{15}+k(2n-1)x^{18}+2knx^{21}+3kn^{2}x^{32}\)
  6. \(M_{a,b}(L(CNC_{k}[n]),x)=2kx^{(2a+3b)}+k(2n-1)x^{3(a+b)}+2knx^{(3a+4b)}+3kn^{2}x^{4(a+b)}\)
  7. \(M^{'}_{a,b}(L(CNC_{k}[n]),x)=2kx^{(a+2)(b+3)}+k(2n-1)x^{(a+2)(b+3)}+2knx^{(a+2)(b+3)}+3kn^{2}x^{(a+2)(b+3)}\)

Proof. Let \(L(CNC_{k}[n])\) be the line graph of Carbon Nanocones \(CNC_{k}[n]\). From the graph of \(CNC_{k}[n]\)(figure 4) we can see that the total number of vertices are \(8k+2kn\) and total number of edges are \(k(n+1)(3n+1)\). The edge set of \(L(CNC_{k}[n])\) has following four partitions $$E_{1}=E_{\{2,3\}}=\{e=uv\in L(CNC_{k}[n]): d_{u}=2, d_{v}=3\}, $$ $$E_{2}=E_{\{3,3\}}=\{e=uv\in L(CNC_{k}[n]): d_{u}=3, d_{v}=3\}, $$ $$E_{3}=E_{\{3,4\}}=\{e=uv\in L(CNC_{k}[n]): d_{u}=3, d_{v}=4\}, $$ and $$E_{4}=E_{\{4,4\}}=\{e=uv\in L(CNC_{k}[n]): d_{u}=4, d_{v}=4\}. $$ Now, $$|E_{1}(L(CNC_{k}[n]))|=2k, $$ $$|E_{2}(L(CNC_{k}[n]))|=k(2n-1), $$ $$|E_{3}(L(CNC_{k}[n]))|=2kn, $$ and $$|E_{4}(L(CNC_{k}[n]))|=3kn^{2}. $$ \begin{eqnarray*} M_{1}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{d_{u}+d_{v}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{2+3}+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3+3}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{3+4}+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4+4}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{5}+|E_{2}(L(CNC_{k}[n]))|x^{6}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{7}+|E_{4}(L(CNC_{k}[n]))|x^{8}\\ &=&2kx^{5}+k(2n-1)x^{6}+2knx^{7}+3kn^{2}x^{8}. \end{eqnarray*} \begin{eqnarray*} M_{2}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{d_{u}\times d_{v}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{2\times 3}+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3\times 3}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{3\times 4}+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4\times 4}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{6}+|E_{2}(L(CNC_{k}[n]))|x^{9}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{12}+|E_{4}(L(CNC_{k}[n]))|x^{16}\\ &=&2kx^{6}+k(2n-1)x^{9}+2knx^{12}+3kn^{2}x^{16}. \end{eqnarray*} \begin{eqnarray*} M_{3}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{d_{v}-d_{u}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{3-2}+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3-3}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{4-3}+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4-4}\\ &=&|E_{1}(L(CNC_{k}[n]))|x+|E_{2}(L(CNC_{k}[n]))|\\ &&+|E_{3}(L(CNC_{k}[n]))|x+|E_{4}(L(CNC_{k}[n]))|\\ &=&k[3n^2+2n-1]+2k(n+1)x. \end{eqnarray*} \begin{eqnarray*} M_{4}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{d_{u}(d_{u}+d_{v})}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{2(2+3)}+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3(3+3)}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{3(3+4)}+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4(4+4)}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{10}+|E_{2}(L(CNC_{k}[n]))|x^{18}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{21}+|E_{4}(L(CNC_{k}[n]))|x^{32}\\ &=&2kx^{10}+k(2n-1)x^{18}+2knx^{21}+3kn^{2}x^{32}. \end{eqnarray*} \begin{eqnarray*} M_{5}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{d_{v}(d_{u}+d_{v})}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{3(2+3)}+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3(3+3)}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{4(3+4)}+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4(4+4)}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{15}+|E_{2}(L(CNC_{k}[n]))|x^{18}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{28}+|E_{4}(L(CNC_{k}[n]))|x^{32}\\ &=&2kx^{15}+k(2n-1)x^{18}+2knx^{28}+3kn^{2}x^{32}. \end{eqnarray*} \begin{eqnarray*} M_{a,b}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}x^{ad_{u}+bd_{v}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{2a+3b}\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{3a+3b}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}x^{3a+4b}\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{4a+4b}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{2a+3b}\\ &&+|E_{2}(L(CNC_{k}[n]))|x^{3a+3b}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{3a+3b}\\ &&+|E_{4}(L(CNC_{k}[n]))|x^{4a+4b}\\ &=&2kx^{(2a+3b)}+k(2n-1)x^{3(a+b)}\\ &&+2knx^{(3a+4b)}+3kn^{2}x^{4(a+b)}. \end{eqnarray*} \begin{eqnarray*} M^{'}_{a,b}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(9L(CNC_{k}[n]))}x^{(a+d_{u})+(b+d_{v})}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}x^{(a+2)+(b+3)}\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}x^{(a+3)+(3+b)}\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{(a+3)+(4+b)}\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}x^{(4+a)+(4+b)}\\ &=&|E_{1}(L(CNC_{k}[n]))|x^{(a+2)+(b+3)}\\ &&+|E_{2}(L(CNC_{k}[n]))|x^{(a+3)+(3+b)}\\ &&+|E_{3}(L(CNC_{k}[n]))|x^{(a+3)+(4+b)}\\ &&+|E_{4}(L(CNC_{k}[n]))|x^{(4+a)+(4+b)}\\ &=&2kx^{(a+2)(b+3)}+k(2n-1)x^{(a+2)(b+3)}\\ &&+2knx^{(a+2)(b+3)}+3kn^{2}x^{(a+2)(b+3)}. \end{eqnarray*}

Theorem 2.2 Let \(L(CNC_{k}[n])\) be the line graph of Carbon Nanocones \(CNC_{k}[n]\). Then,

  1. Re\(ZG_{1}(L(CNC_{k}[n]))=\frac{3}{2}kn^{2}+\frac{5}{2}kn+k\),
  2. Re\(ZG_{2}(L(CNC_{k}[n]))=6kkn^{2}+\frac{45}{7}kn+\frac{9}{10}k\),
  3. Re\(ZG_{3}(L(CNC_{k}[n]))=6k(64n^{2}+46n+1)\),

Proof.

    \begin{eqnarray*} ReZG_{1}(L(CNC_{k}[n]))&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}\frac{d_{u}+d_{v}}{d_{u}.d_{v}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}\frac{2+3}{2.3}\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}\frac{3+3}{3.3}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}\frac{3+4}{3.4}\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}\frac{4+4}{4.4}\\ &=&|E_{1}(L(CNC_{k}[n]))|\frac{5}{6}+|E_{2}(L(CNC_{k}[n]))|\frac{6}{9}\\ &&+|E_{3}(L(CNC_{k}[n]))|\frac{7}{12}+|E_{4}(L(CNC_{k}[n]))|\frac{8}{16}\\ &=&(2k)\frac{5}{6}+k(2n-1)\frac{6}{9}+(2kn)\frac{7}{12}+(3kn^{2})\frac{1}{2}\\ &=&\frac{5}{3}k+\frac{4}{3}kn-\frac{2}{3}k+\frac{7}{6}kn+\frac{1}{2}kn^{2}\\ &=&\frac{3}{2}kn^{2}+\left(\frac{24+21}{18}\right)kn+\left(\frac{5-2}{3}\right)k\\ &=&\frac{3}{2}kn^{2}+\frac{5}{2}kn+k. \end{eqnarray*} \begin{eqnarray*} ReZG_{2}(L(CNC_{k}[n]),x)&=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}\frac{d_{u}.d_{v}}{d_{u}+d_{v}}\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}\frac{2.3}{2+3}\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}\frac{3.3}{3+3}\\ &&+\sum\limits_{uv\in E_{3}(L(CNC_{k}[n]))}\frac{3.4}{3+4}\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}\frac{4.4}{4+4}\\ &=&|E_{1}(L(CNC_{k}[n]))|\frac{6}{5}+|E_{2}(L(CNC_{k}[n]))|\frac{9}{6}\\ &&+|E_{3}(L(CNC_{k}[n]))|\frac{12}{7}+|E_{4}(L(CNC_{k}[n]))|\frac{16}{8}\\ &=&(2k)\frac{6}{5}+k(2n-1)\frac{9}{6}+(2kn)\frac{12}{6}+2(3kn^{2})\\ &=&\frac{12}{5}k+3kn-\frac{3}{2}k+\frac{24}{7}kn+6kn^{2}\\ &=&6kn^{2}+\left(\frac{24+21}{7}\right)kn+\left(\frac{24-15}{10}\right)k\\ &=&6kkn^{2}+\frac{45}{7}kn+\frac{9}{10}k. \end{eqnarray*} \begin{eqnarray*} ReZG_{3}(L(CNC_{k}[n]),x) &=& \sum\limits_{uv\in E(L(CNC_{k}[n]))}(d_{u}.d_{v})(d_{u}+d_{v})\\ &=& \sum\limits_{uv\in E_{1}(L(CNC_{k}[n]))}(2.3)(2+3)\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}(3.3)(3+3)\\ &&+\sum\limits_{uv\in E_{2}(L(CNC_{k}[n]))}(3.4)(3+4)\\ &&+\sum\limits_{uv\in E_{4}(L(CNC_{k}[n]))}(4.4)(4+4)\\ &=&|E_{1}(L(CNC_{k}[n]))|30+|E_{2}(L(CNC_{k}[n]))|54\\ &&+|E_{3}(L(CNC_{k}[n]))|84+|E_{4}(L(CNC_{k}[n]))|128\\ &=&(2k)\frac{5}{6}+k(2n-1)\frac{6}{9}+(2kn)\frac{7}{12}+(3kn^{2})\frac{1}{2}\\ &=&30(2k)+54k(2n-1)+84(2kn)+128(3kn^{2})\\ &=&60k+108kn-54k+168kn+384kn^{2}\\ &=&6k(64n^{2}+46n+1). \end{eqnarray*}

Competing Interests

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

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Category: OMA - Vol 2 - Issue 1 (2018)