Advances in Physical Chemistry

Volume 2016 (2016), Article ID 2315949, 6 pages

http://dx.doi.org/10.1155/2016/2315949

## General Randić, Sum-Connectivity, Hyper-Zagreb and Harmonic Indices, and Harmonic Polynomial of Molecular Graphs

^{1}Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran 16844, Iran^{2}School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China^{3}Department of Mathematics, Maharani’s Science College for Women, Mysore 570005, India^{4}Department of Mathematics, The National Institute of Engineering, Mysuru 570008, India^{5}School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China

Received 7 July 2016; Accepted 30 August 2016

Academic Editor: Dennis Salahub

Copyright © 2016 Mohammad Reza Farahani et al. 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.

#### Abstract

We present explicit formula for the general Randić connectivity, general sum-connectivity, Hyper-Zagreb and Harmonic Indices, and Harmonic polynomial of some simple connected molecular graphs.

#### 1. Introduction

In this paper, we consider only simple connected graphs without loops and multiple edges. A connected graph is a graph such that there is a path between all pairs of vertices. Let be an arbitrary simple connected graph; we denote the vertex set and the edge set of by and , respectively. For two vertices and of , the distance between and is denoted by and defined as the length of any shortest path connecting and in . For a vertex of , the degree of is denoted by and is the number of vertices of adjacent to .

In chemical graph theory, we have many invariant polynomials and topological indices for a molecular graph. A topological index is a numerical value for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity [1–3].

One of the oldest topological indices or molecular descriptors is the Zagreb index that has been introduced more than forty years ago by Gutman and Trinajstić in 1972 [4].

Now, we know that, for a molecular graph , the first Zagreb index and the second Zagreb index are defined as

Recently, a new version of Zagreb indices named Hyper-Zagreb index was introduced by Shirdel et al. in 2013 [5] and it is defined as

We encourage the reader to consult [6–30] for historical background and mathematical properties of the Zagreb indices.

In 1975, Randić proposed a structural descriptor called the branching index [31] that later became the well-known Randić molecular connectivity index. Motivated by the definition of Randić connectivity index based on the end-vertex degrees of edges in a graph defined as the sum of the weights of all edges of ,

Later, the Randić connectivity index had been extended as the general Randić connectivity index, which is defined as the sum of the weights and is equal to

Also, a closely related variant of Randić connectivity index called the sum-connectivity index was introduced by Zhou and Trinajstić in 2008 [32, 33]. The sum-connectivity index is defined as

The general sum-connectivity index of a graph is equal to

In 1987 [34], Fajtlowicz introduced the Harmonic index of a graph which is defined as the sum of the weights of and is equal to

The Harmonic index is one of the most important indices in chemical and mathematical fields. It is a variant of the Randić index which is the most successful molecular descriptor in structure-property and structure activity relationships studies. The Harmonic index gives somewhat better correlations with physical and chemical properties compared with the well-known Randić index. Estimating bounds for is of great interest, and many results have been obtained. For example, Favaron et al. [35] considered the relationship between the Harmonic index and the eigenvalues of graphs, and Zhong [36–38] determined the minimum and maximum values of the Harmonic index for simple connected graphs, trees, unicyclic graphs, and bicyclic graphs and characterized the corresponding extremal graphs, respectively. It turns out that trees with maximum and minimum Harmonic index are the path and the star , respectively.

Recently, Iranmanesh and Salehi [39] introduced the Harmonic polynomial of a graph which is equal towhere .

We encourage the reader to consult [40–43] for more history and mathematical properties of the Randić index and the Harmonic index.

In this paper, we present explicit formula for the general Randić connectivity, general sum-connectivity, Hyper-Zagreb and Harmonic Indices, and Harmonic polynomial of some hydrocarbon molecular graphs.

#### 2. Results and Discussion

In this section, we compute the general Randić connectivity, general sum-connectivity indices, the Hyper-Zagreb and Harmonic Indices, and Harmonic polynomial of a family of hydrocarbon molecules, which are called Polycyclic Aromatic Hydrocarbons .

The Polycyclic Aromatic Hydrocarbons is a family of hydrocarbon molecules, such that its structure is consisting of cycles with length six (benzene). The Polycyclic Aromatic Hydrocarbons can be thought of as small pieces of graphene sheets with the free valences of the dangling bonds saturated by . Vice versa, a graphene sheet can be interpreted as an infinite PAH molecule. Successful utilization of PAH molecules in modeling graphite surfaces has been reported earlier [44–52] and references therein. Some first members and a general representation of this hydrocarbon molecular family are shown in Figures 1 and 2.