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 .
We assume that the probability is a constant, invariant to the step parameter . That is, the process described is a zeroth-order Markov process.
By eliminating, “squeezing out,” the squares from a phenylene, a catacondensed hexagonal system (which may be jammed) is obtained, called the hexagonal squeeze of the respective phenylene (see Figure 4). Clearly, there is a one-to-one correspondence between a phenylene (PH) and its hexagonal squeeze (HS). Both possess the same number of hexagons. The respective hexagonal squeeze of a random phenylene chain is a random hexagonal chain, and we denote it by .
The Wiener index and the number of perfect matchings of a random hexagonal chain have been studied by Gutman et al. [8–10]. Analogous results for a random phenylene chain have been obtained by Chen and Zhang in [11]. In this paper we obtain explicit expressions for the expected value of the Merrifield-Simmons index of a random phenylene chain and a random hexagonal chain , respectively. We also compute the corresponding entropy constants and obtain the maximum and minimum values in both random systems, respectively. For some recent results on these hexagonal structures, see [12, 13].
2. Merrifield-Simmons Index of a Random Phenylene Chain
Firstly, let us recall some results in [14], useful to this paper.
Lemma 1 (see [14]). Consider .
Lemma 2 (see [14]). Let be a vertex of and let be the subset of consisting of the vertex and its neighbors. Then
As described above, the phenylene chain can be obtained by adjoining to a hexagon by a 4-membered ring. For this construction, the following relations are easily obtained by Lemmas 1 and 2.
Lemma 3. Let be denoted as a phenylene chain as in Figure 2; then for one has
Proof. Consider By the symmetry, For a random phenylene chain , the Merrifield-Simmons indices , , and are random variables and we denote their expected values by , , and , respectively.
By the definition of , we immediately have that, for ,By the symmetry, To solve the recursion equation, we use the method of the generating functions. Set Then we have that Recall that , , and .
Solving the equations, we have that where
So we have the following result.
Theorem 4. If , then for one has that where Specifically, one has that
The following corollary is easily obtained from Theorem 4 which gives the limits of the entropy constant as , where is the vertex set of .
Corollary 5. If , then for , one has It is easy to check that is a monotonic decreasing function on , so the limit of has the maximum value at and the minimum value at . That is say, for different (), the limit of has little difference.
3. Merrifield-Simmons Index of a Random Hexagon Chain
Similar to the phenylene chain , the hexagon chain can be obtained by adjoining to a hexagon. For this construction the following relations are easily obtained by Lemmas 1 and 2.
Lemma 6. Let be denoted as a hexagon squeeze of a phenylene chain as in Figure 2; then for one has
Proof. Consider By the symmetry,For a random hexagon chain , the Merrifield-Simmons indices , , , and are also random variables, and in not confusion circumstances we also denote their expected values by , , , and , respectively. Then we immediately have that, for , By the symmetry, Just as above, we set Then we have that Recall that , , and .
Solving the equations, we have thatwhere
So we have the following result.
Theorem 7. If , then for one has that whereSpecifically, we have that
The following corollary is easily obtained from Theorem 7 which gives the limit of as , where is the vertex set of .
Corollary 8. If , then for , one has It is easy to check that is a monotonic decreasing function on , so the limit of has the maximum value at , and the minimum value at . That is say, for different , the limit of has little difference.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the National Natural Science Foundations of China (no. 11401102).