Abstract
A perfect matching of a (molecule) graph is a set of independent edges covering all vertices in . In this paper, we establish a simple formula for the expected value of the number of perfect matchings in random octagonal chain graphs and present the asymptotic behavior of the expectation.
1. Introduction
A general problem of interest in chemistry, physics, and mathematics is the enumeration of perfect matchings, on lattices and (molecule) graphs. Let be a graph. A perfect matching of is a set of independent edges covering all vertices in , which is called Kekulé structure in organic chemistry and closed-packed dimer in statistical physics. In organic chemistry, there are strong connections between the number of the Kekulé structures and chemical properties for many molecules such as benzenoid hydrocarbons [1–3]. The number of Kekulé structures is an important topological index which had been applied for estimation of the resonant energy and total -electron energy [2, 4] and Clar aromatic sextet [5]. In crystal physics, the perfect matching problem is closely related to the dimer problem [6–8]. Denote the number of perfect matchings of a graph by .
An octagonal system (graph) [9] is a finite 2-connected geometric graph in which every interior face is bounded by a regular octagon or quadrangle of side length 1 (see Figure 1). Octagonal graphs have attracted many mathematicians’ considerable attentions because of many interest combinatorial subjects. Brunvoll et al. [9] determined the number of isomers of tree-like octagonal graphs by the generating functions. Su and Ding [10] showed that there is a side whose relative length is at most 1 in every convex octagon, and this bound is asymptotically tight. In 2001, Destainville et al. [11] considered some combinatorial properties of fixed-boundary octagonal random tilings. Yang and Zhao [12] presented a relation between the number of perfect matching in octagonal chain graphs and Hosoya index of the caterpillar trees in 2013. Wei et al. [13]discussed the Wiener indices in a type of random octagonal chains in 2018.
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An octagonal graph, called an octagonal chain graph, proved that no octagon is adjacent to more than two other octagons. Both the octagonal graphs and are octagonal chain graphs, as shown in Figure 1. Let be an octagonal chain graph with octagons labeled by , where and are adjacent for each (). Both the first octagon and the last octagon are called terminal octagon. And, the remaining octagons are called internal octagon. Each internal octagon is one of type , type , or type according to whether it separates its two adjacent octagons by a distance of 3, 1, or 2, as shown in Figure 2. A random octagonal chain graph of length is an octagonal chain graph with octagons in which each internal octagon is one of type with probability , type with probability , or type with probability , denoted by . Gutman [14, 15] studied the perfect matchings about random benzenoid chain graphs in 1990s. Chen and Zhang [16] obtained a simple exact formula for the expected value of the number of perfect matchings in a random phenylene chain. Simple exact formulae are presented for the expected value of the number of perfect matchings in random polyomino chain graphs by Wei et al. [17] in 2016. Recently, Wei and Shiu [18] obtained the expected value of the number of perfect matchings in random polyazulenoid chains.
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In this paper, we establish an exact formula for the expected value of the number of perfect matchings in a random octagonal chain graph.
2. The Number of Perfect Matchings in Random Octagonal Chain Graphs
In this section, we consider the expected value of the number of perfect matchings in a random octagonal chain graph. We will keep the notation defined in Section 1. Recall that there is a recursive formula for the perfect matchings in [4], i.e.,where denotes an edge of incident with the vertices and . All notations which are not defined in this paper can be found in [19].
Lemma 1. Let be an octagonal chain graph with octagons. Then,and for ,
Proof. Without loss of generality, let be an edge in the octagonal chain graph , as shown in Figures 3 and 4. Case 1. Suppose the th octagon is of type or . It is easy to see from Figure 3 that Thus, we get the result by (1), i.e., Case 2. Suppose the th octagon is of type . It is easy to see from Figure 4 thatThus, we get the result by (1), i.e.,This completes the proof.
Note that the probabilities and are unknown constants. Here, is a random variable. Denote the expected value of by .
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Lemma 2. Let be a random octagonal chain graph with octagons. Then,where .
Proof. Since the th octagon of is one of type with probability , type with probability , and type with probability , we haveby Lemma 1. Recall that . Since is a sum of random variables, we haveThus, the proof is completed.
Theorem 1. Let be a random octagonal chain graph with octagons.(1)If , then, for each , where(2)If , then
Proof. Let and . Since and , we have the initial conditions , . By Lemma 2, we havefor . Note that the characteristic equation of (14) isand its characteristic roots are Case 1. If , then . In this case, Substituting the boundary conditions and , we obtain Therefore, which proves the first statement of the theorem. Case 2. If , then . Thus, , and we havewhich means that .
Let , , and be the parachain, orthochain, and metachain with octagons, as shown in Figure 5. By assuming , , and , respectively, we can obtain the number of Kekulé structures of , , and from Theorem 1.
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Corollary 1. Let , , and be the parachain, orthochain, and metachain with octagons. Then,(1)(2)(3)It is suggested that the function has interest in mathematics and chemistry in [14], especially concerning its asymptotic behavior with respect to . From the explicit expression for in Theorem 1, we have the following result.
Corollary 2. Let be a random cyclooctane chain graph with octagons. If , thenwhere
Proof. If , thenby Theorem 1. And, by the explicit expression of , we haveSince , we obtainThus, the proof is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no potential conflicts of interest.
Acknowledgments
This work was supported by National Natural Science Foundation of China (no. 12061007), Natural Science Found of Fujian Province (no. 2020J01844), and Science Foundation for the Education Department of Fujian Province (no. JT180392).