#### SOME CONNECTIVITY INDICES AND ZAGREB INDEX OF HONEYCOMB GRAPHS

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**SOME CONNECTIVITY INDICES AND ZAGREB INDEX OF HONEYCOMB GRAPHS**

Yingying Gao ^{1}, Muhammad Imran ^{2}, Mohammad Reza Farahani^{*3} and Hafiz Muhammad Afzal Siddiqui ^{4}

School of Biological Science ^{1}, Guangzhou University, Guangzhou 510006, China.

Department of Mathematical Sciences ^{2}, United Arab Emirates University, P.O. Box 15551, Al Ain, United Arab Emirates.

Department of Mathematics ^{2}, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-12, Islamabad, Pakistan.

Department of Applied Mathematics ^{3}, Iran University of Science and Technology, (IUST) Narmak, Tehran 16844, Iran.

Department of Mathematics ^{4}, COMSATS Institute of Information Technology, Lahore, Pakistan.

**ABSTRACT:** In this paper, we investigate several topological indices in honeycomb graphs: Randić connectivity index, sum-connectivity index, atom-bond connectivity index, geometric-arithmetic index, First and Second Zagreb indices and Zagreb polynomials. Formulas for computing the above topological descriptors in honeycomb graphs are given.

Keywords: |

Topological index, Honeycomb graph, Atom-bond connectivity index, Randić connectivity index

**INTRODUCTION: **In this paper, we only consider an undirected graph without loops and multiple edges. Let *G* be a graph, we denote *V (G)* and *E(G)* the vertex set and edge set of G, respectively. An edge uv is an (*s; t*)-edge if *d(u) *= *s* and *d(v) *= *t*. In chemical graph theory, the vertices of the graph correspond to the atoms of molecules and the edges correspond to chemical bonds. The Wiener index *W (G)* is defined as the sum of topological distances *d*(*u; v*) between any two atoms in the molecular graph, which is introduced by the chemist Harold Wiener ^{1}.

^{}

According to the above Zagreb indices, the First Zagreb polynomial *Zg _{1}(G; x)* and the Second Zagreb polynomial

*Zg*have been defined. They are defined as

_{2}(G; x)**Honeycomb Graph: **The *n*-honeycomb graph *Hn* was studied in ^{10}. It is the graph shown in **Fig. 1**, where mean of vertex is vertex of the hexagons and mean of edges are the sides of the hexagons.

**Definition 1: **For any integer. Let *P _{1}*;

*P*;…;

_{2}*P*be

_{k}*k*paths such that:

**FIG. 1: THE n-HONEYCOMB GRAPH Hn WITH 1 ≤ n ≤ 10**

**RESULT:**

**Lemma 1:**

**Proof:**

**A B**

**FIG. 2: (A) THE VERTICES IN H_{5}; (B) THE VERTICES IN H_{6}**

**Theorem 1:** If *n* is even, then

**Proof:**

**Theorem 2:** If n is odd, then

**Proof:**

**CONCLUSION: **In this paper, we computed topological indices “Randić connectivity index, sum-connectivity index, atom-bond connectivity index, geometric-arithmetic index, First and Second Zagreb indices and Zagreb polynomials” of

a honeycomb graph Hk. Formulas for computing the above topological descriptors in honeycomb graphs are given and such results of other types of chemical graphs.

**ACKNOWLEDGEMENT: **This work was supported by applied basic research (Key Project) of Sichuan Province under grant 2017JY0095, key project of Sichuan provincial department of education under grant 17ZA0079 and automotive creative design pilot area of chengdu University and Longquanyi District under grant 2015-CX00-00010-ZF and higher education commission of Pakistan *via* Ref. No. 20-367/NRPU/R D/HEC/ 12/831 and by National University of Sciences and Technology, Islamabad, Pakistan.

**CONFLICT OF INTEREST: **The authors declare that there is no conflict of interests regarding the publication of this paper.

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**How to cite this article:**Gao Y, Imran M, Farahani MR and Siddiqui HMA: Some connectivity indices and zagreb index of honeycomb graphs. Int J Pharm Sci Res 2018; 9(5): 2080-87.doi: 10.13040/IJPSR.0975-8232.9(5).2080-87.

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# Article Information

48

2080-2087

852

828

English

IJPSR

Y. Gao, M. Imran, M. R. Farahani * and H. M. A. Siddiqui

Department of Applied Mathematics, Iran University of Science and Technology, (IUST) Narmak, Tehran, Iran.

Mrfarahani88@gmail.com

12 August, 2017

17 October, 2017

20 October, 2017

10.13040/IJPSR.0975-8232.9(5).2080-87

01 May, 2018