Abstract
The polyphenyl system is composed of hexagons obtained from two adjacent hexagons that are sticked by a path with two vertices. The Hosoya index of a graph is defined as the total number of the independent edge sets of . In this paper, we give a computing formula of Hosoya index of a type of polyphenyl system. Furthermore, we characterize the extremal Hosoya index of the type of polyphenyl system.
1. Introduction
The polyphenyl system is composed of hexagons obtained from two adjacent hexagons that are sticked by a path . Polyphenyl systems are of great importance for theoretical chemistry because they are natural molecular graph representations of benzenoid hydrocarbons [1]. Polyphenyl systems are graph representations of an important subclass of benzenoid molecules.
A topological index is a numerical quantity derived in a unambiguous manner from the structure graph of a molecule. As a graph structural invariant, it does not depend on the label or the pictorial representation of that graph. Various topological indices usually reflect the molecular size and shape. One topological index is Hosoya index, which was first introduced by Hosoya [2]. It plays an important role in the so-called inverse structure-property relationship problems. For details of Hosoya index and its applications, the readers are suggested to refer to [1, 3–5] and references therein. For other topological indices, please see [6–23], among others.
In this paper, our aim is to find the computation formula of Hosoya index of a polyphenyl system. We present some definitions and notations as follows.
Let be a graph with vertex set and edge set . Let and be an edge and a vertex of , respectively. We will denote by or the graph obtained from by removing or , respectively. Denote by the set . Let be a subset of . The subgraph of induced by is denoted by , and is denoted by .
Two edges of are said to be independent if they are not adjacent in . A -matching of is a set of mutually independent edges. Denote by the number of the -matchings of . For convenience, let for any graph . The Hosoya index of , denoted by , is defined as where stands for the order, the number of vertices, of and is the integer part of .
We denote by hexagonal degree of a polyphenyl system graph, which is the number of hexagons sticked by three s. A polyphenyl system graph is called the polyphenyl spider (see Figure 1) if and called polyphenyl chain if . Let polyphenyl chain be composed of hexagons obtained from two vertices of adjacent hexagons and that are vertex-sticked by two end vertices of path , respectively. If the two vertex sets of in divided by two path s both have two vertices, then it is called a linear polyphenyl chain, denoted by . Let , and be induced subgraph of with hexagons obtained by deleting some vertices (see Figures 2 and 3).
We denote by the set of polyphenyl spiders with hexagons. A polyphenyl spider is called a -type polyphenyl spider and denote where if three branches of after deleting the hexagon which sticked by three paths are linear polyphenyl chains.
2. Some Lemmas
In this section, we will give some lemmas which will be used later.
Lemma 1 (see [1]). Let be a graph consisting of two components and . Then
Lemma 2 (see [1]). Let be a graph and any . Then
By Lemmas 1 and 2, we can obtain the following two results.
Lemma 3. Let be a linear polyphenyl chain with hexagons and , , and be three chains with hexagons. Then(i),(ii),(iii).
Lemma 4. Let be a -type polyphenyl spider with hexagons. Then
Lemma 5 (see [24]). Let and be a Fibonacci and Lucas sequences, respectively. Then(i), , where and ,(ii),(iii).
Lemma 6 (see [25]). Let be all different roots of the homogeneous recursive formula . And let be the multiplicity of . Then the general solution of homogeneous recursive formula is , where for .
Lemma 7 (see [25]). Let be the nonhomogeneous recursive formula, where and are constants. If is the general solution of homogeneous recursive formula , then the general solution of the above nonhomogeneous recursive formula can be expressed as , where and are fixed constants.
Lemma 8 (see [26]). Let be a linear polyphenyl chain with hexagons. Then
Furthermore, Lemma 8 also can be expressed as another form, that is, the following lemma.
Lemma 9 (see [26]). Let be the linear polyphenyl chain with hexagons. Then
3. Main Results
Theorem 10. Let be a linear polyphenyl chain with hexagons and , , and be three chains with hexagons. Then(i),(ii),(iii).
Proof. Combining Lemmas 3 and 8 and (i) of Theorem 10, it is easy to prove (ii) and (iii) of Theorem 10. We only prove (i) of Theorem 10 as follows.
By Lemma 3, we have By Lemma 9, we get that By Lemma 6, solving the homogeneous recursive formula of (8), we obtain that . By Lemma 7, the general solution of the nonhomogeneous recursive formula (8) can be expressed as where and are fixed constants. For the sake of simplicity, we set and . Then the general solution of the nonhomogeneous recursive formula (9) can be expressed as where , , are fixed constants. Substituting (10) into the nonhomogeneous recursive formula (8), we get that By direct calculation, we get . By (12), we have . And the proof of Theorem 10 is complete.
Theorem 11. Let be the -type polyphenyl spider with hexagons. Then
Proof. For the sake of facilitating the calculation, we set the coefficients of all formulas of Lemma 8 and Theorem 10 as follows:By Lemma 4, we know that By Lemma 9 and Theorem 10, simplifying (15), we have
By Theorem 11, we can obtain two corollaries as follows.
Corollary 12. Let be a -type polyphenyl spider with hexagons. Then Particularly, the equality holds if and only if
Corollary 13. Let be the -type polyphenyl spider with hexagons. Then where the equality holds if and only if .
4. Conclusion
In this paper, applying the relation between inhomogeneous constant coefficient recursion formula and constant coefficient recursion formula, we give a computing formula of Hosoya index of a -type polyphenyl spider. Furthermore, we determine completely the -type polyphenyl spider which has the largest and smallest Hosoya index.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by NSFC (11502132), NSF of Qinghai (2016-ZJ-947Q), education department of Shaanxi province (15JK1135), and Shaanxi Sci-Tech University (SLGQD14-14).