Table of Contents
Advances in Physical Chemistry
Volume 2016, Article ID 2315949, 6 pages
http://dx.doi.org/10.1155/2016/2315949
Research Article

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

1Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran 16844, Iran
2School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
3Department of Mathematics, Maharani’s Science College for Women, Mysore 570005, India
4Department of Mathematics, The National Institute of Engineering, Mysuru 570008, India
5School 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 [13].

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 [630] 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 [3638] 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 [4043] 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 [4452] and references therein. Some first members and a general representation of this hydrocarbon molecular family are shown in Figures 1 and 2.

Figure 1: Some first members of the Polycyclic Aromatic Hydrocarbons ().
Figure 2: A general representation of the hydrocarbon molecular family “Polycyclic Aromatic Hydrocarbons, ”.

Theorem 1 (see [45]). Consider the Polycyclic Aromatic Hydrocarbons . Then, the first and second Zagreb indices of are equal to

Theorem 2. The Hyper-Zagreb index of Polycyclic Aromatic Hydrocarbons is equal to

Theorem 3 (see [46]). The Randić connectivity and sum-connectivity indices of the Polycyclic Aromatic Hydrocarbons are equal to

Theorem 4. Let be the Polycyclic Aromatic Hydrocarbons. Then, (i)the general Randić connectivity index of is equal to(ii)the general sum-connectivity index of is equal to

Theorem 5. Consider the Polycyclic Aromatic Hydrocarbons . Then,(i)the Harmonic index of is equal to :(ii)the Harmonic polynomial of is equal to :

Before presenting the main results, consider the following definition.

Definition 6 (see [10]). Let be a simple connected molecular graph. We divide the vertex set and edge set of based on the degrees of a vertex/atom in . Obviously, and we denote the minimum and maximum of the by and , respectively:

Proof of Theorem 2. Let be the Polycyclic Aromatic Hydrocarbon for all integer numbers . From the general representation of in Figure 2, one can see that in this hydrocarbon molecular family there are vertices/atoms such that of them are carbon atoms and also of them are hydrogen atoms. In other words,Thus, there are edges/chemical bonds in .
Now, by using Definition 6 and according to Figure 2, one can see that in hydrocarbon molecules all hydrogen atoms have one connection and the degree of them is and there are 3 edges/chemical bonds for all carbon atoms; thus, .
Therefore, we have two partitions of the vertex set of Polycyclic Aromatic.
Hydrocarbons are as follows:On the other hand, from Figure 2 and [45, 46], we can see that and .
Here, we have the following computations for the Hyper-Zagreb index of the Polycyclic Aromatic Hydrocarbons as follows:Here, we complete the proof of Theorem 2.

Proof of Theorem 4. Consider the Polycyclic Aromatic Hydrocarbons with vertices/atoms and edges. Then, by using the results from the above proof, we have the following computations for the general Randić and sum-connectivity indices of :

Proof of Theorem 5. Let be the Polycyclic Aromatic Hydrocarbon for all integer numbers . By results from proof of Theorem 2, we see that the Harmonic index and Harmonic polynomial of are equal toHere, the proof of Theorem 5 was completed.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are thankful to Professor Mircea V. Diudea, Faculty of Chemistry and Chemical Engineering, Babes-Bolyai University, for his precious support and suggestions.

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