Discrete Dynamics in Nature and Society

Volume 2015 (2015), Article ID 568926, 7 pages

http://dx.doi.org/10.1155/2015/568926

## Merrifield-Simmons Index in Random Phenylene Chains and Random Hexagon Chains

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 31 January 2015; Accepted 13 March 2015

Academic Editor: Alicia Cordero

Copyright © 2015 Ailian Chen. 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 author obtains explicit expressions for the expected value of the Merrifield-Simmons index of a random phenylene chain and a random hexagon chain, respectively. The author also computes the corresponding entropy constants and obtains the maximum and minimum values in both random systems, respectively.

#### 1. Introduction

Let be a simple undirected graph on vertices. Two vertices of are said to be independent if they are not adjacent in . A -independent set of is a set of mutually independent vertices. Denote by the number of the -independent sets of . By definition, the empty vertex set is an independent set. Then for any graph . The Merrifield-Simmons index of , denoted by , is defined as . So is equal to the total number of the independent sets of . The Merrifield-Simmons index was introduced in 1982 by Prodinger and Tichy [1], where it was called the Fibonacci number of a graph. The Merrifield-Simmons index is one of the most popular topological indices in chemistry intensively studied, as seen in the monograph [2]. Recently, there have been many papers studying the Merrifield-Simmons index for a graph. For more details see [3–8], among others.

Phenylenes are a class of conjugated hydrocarbons composed of six- and four-membered rings, where the six-membered rings (hexagons) are adjacent only to four-membered rings, and every four-membered ring is adjacent to a pair of nonadjacent hexagons. If each six-membered ring of a phenylene is adjacent only to two four-membered rings, we say that is a phenylene chain. Due to their aromatic and antiaromatic rings, phenylenes exhibit unique physicochemical properties. In Figure 1, some examples of phenylene chains are presented. The unique phenylene chains for and are shown in Figure 1. More generally, a phenylene chain with hexagons (see Figure 2) can be regarded as a phenylene chain , with hexagons to which a new terminal hexagon has been adjoined by a four-membered ring. But, for , the terminal hexagon can be attached in three ways, which results in the local arrangements we describe as , , and (see Figure 3). Naturally, we define a random phenylene chain with hexagons as a phenylene chain obtained by stepwise addition of terminal hexagons. At each step (=) a random selection is made from one of the three possible constructions: (1) , with probability , (2) , with probability , or (3) , with probability .