Discrete Dynamics in Nature and Society

Volume 2018, Article ID 5702346, 5 pages

https://doi.org/10.1155/2018/5702346

## The General ()-Path Connectivity Indices of Polycyclic Aromatic Hydrocarbons

School of Science, China University of Geosciences (Beijing), Beijing, 100083, China

Correspondence should be addressed to Haiying Wang; moc.621@thcyhw

Received 15 June 2018; Accepted 26 August 2018; Published 5 September 2018

Academic Editor: Luisa Di Paola

Copyright © 2018 Haiying Wang and Chuantao Li. 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

The general -path connectivity index of a molecular graph originates from many practical problems such as three-dimensional quantitative structure-activity (3D QSAR) and molecular chirality. It is defined as , where the summation is taken over all possible paths of length of and we do not distinguish between the paths and . In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons (), which play a role in organic materials and medical sciences. We try to compute the exact general -path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of for any real number . These valuable results could produce strong guiding significance to these applied sciences.

#### 1. Introduction

##### 1.1. Application Background

In many fields like physics, chemistry, and electric network, the boiling point, the melting point, the chemical bonds, and the bond energy are all important quantifiable parameters in their fields.

To understand physicochemical properties of chemical compounds or network structures, we abstractly define different concepts, collectively named* topological descriptors* or* topological indices* after mathematical modelings. We called them different names such as Randić index and Zagreb index. Different index represents its corresponding chemical structures in graph-theoretical terms via arbitrary molecular graph. A large number of articles about related all* topological indices* are proposed and based on edges or vertices in molecular graph [1–3].

In the last decades, as a powerful approach, these two-dimensional topological indices have been used to discover many new drugs such as Anticonvulsants, Anineoplastics, Antimalarials or Antiallergics, and Silico generation ([4–8]). Therefore, the practice has proven that the topological indices and quantitative structure-activity relationships (QSAR) have moved from an attractive possibility to representing a foundation stone in the process of drug discovery and other research areas ([9–12]).

Most importantly, with the further study of chemical indices and drug design and discovery, three-dimensional molecular features (topographic indices) and molecular chirality are also presented. It is more and more urgent to study the three-dimensional quantitative structure-activity (3D QSAR) such as molecular chirality. Actually, so far there have been few results expect that one related definition which is generally mentioned in [7].

##### 1.2. Notations

Throughout this paper, we always let be a simple molecular graph with the vertex set and the edge set . Denote the numbers of vertices and edges by and respectively. In physicochemical graph theory, the vertices and the edges correspond to the atoms and the bonds, respectively. Two vertices and are* adjacent* if there exists an edge between them in . The number of its adjacent vertices is called* degree* of , denoted by or . The set of all of neighbors of is denoted by or . Specially, a vertex in is called* pendant* if its degree is one. All other notations and terminologies are referred to [13].

With the intention of extending the applicability of the general Randić index, L.B.Kier, L.H.Hall, E. Estrada, and coworkers considered* the general **-path connectivity index* of aswhere the summation is taken over all possible paths of length of and we do not distinguish between two paths and ([4, 5]).

According to the definition above, it is clear that the general -path connectivity index of a graph is a real number and an important invariant under graph automorphism. It is closely related to the structures of a molecular graph. For any molecular material, only by mastering its structure can we calculate its exact value of the general -path connectivity index.

In this paper, we focus on the structures of Polycyclic Aromatic Hydrocarbons, for short , which play a role in organic materials and medical science. Then we try to compute the general -path connectivity indices of this family of hydrocarbon structures. Furthermore, we exactly derive the monotonicity and the extremal values of for any real number . The valuable results could produce strong guiding significance to these applied sciences.

For convenience, it is necessary to simplify some basic concepts and notations in . An -vertex denotes a vertex with degree , and a -edge stands for an edge connecting a -vertex with a -vertex. Let denote the number of -edge.

Let be a path of length in , that is, . And , is called its degree sequence. Then it is obvious that there are all three types , , and of degree sequences of -paths in . Let , , and denote the numbers of the -paths of the degree sequence types and and in , respectively.

#### 2. Polycyclic Aromatic Hydrocarbons

Large Polycyclic Aromatic Hydrocarbons are ubiquitous combustion products and belong to more important hydrocarbon molecules. They have been implicated as carcinogens and play a role in organic materials and medical science [14].

As we known, Polycyclic Aromatic Hydrocarbons have great significance as molecular analogues of graphite as candidates for interstellar species and as building blocks of functional materials for device applications. In addition, synthetic routes to Polycyclic Aromatic Hydrocarbons are available. Therefore, much detailed knowledge of all molecular features would be necessary for the tuning of molecular properties towards specific applications.

Polycyclic Aromatic Hydrocarbons can be regarded as graphene sheets composed of free radicals of saturated suspended bonds and vice versa; graphene sheets can be interpreted as an infinite number of PAH molecules. The successful application of Polycyclic Aromatic Hydrocarbons in modeling of graphite surface has been reported and references have been provided. The family of Polycyclic Aromatic Hydrocarbons have similar properties with Benzenoid system (Circumcoronene Homologous Series of Benzenoid) ([9–12]). Thus, molecular structures of Polycyclic Aromatic Hydrocarbons play a key role in particular.

For any positive integer , let be the general representation of this Polycyclic Aromatic Hydrocarbon shown in Figure 1. To understand more structures of , the first three members of this hydrocarbon family are given in Figure 2, where is called Benzene with 6 carbon atoms () and 6 hydrogen () atoms, the Coronene with carbon atoms and hydrogen atoms, Circumcoronene with carbon atoms, and hydrogen atoms.