Journal of Applied Mathematics

Volume 2012 (2012), Article ID 520156, 8 pages

doi:10.1155/2012/520156

Research Article

The Merrifield-Simmons Index and Hosoya Index of Graphs

Shaojun Dai¹ and Ruihai Zhang²

¹Department of Mathematics, Tianjin Polytechnic University, No. 399 Binshuixi Road, Xiqing District, Tianjin 300387, China

²Department of Mathematics, Tianjin University of Science and Technology, Tianjin 300457, China

Received 29 May 2012; Revised 18 June 2012; Accepted 27 June 2012

Academic Editor: Ferenc Hartung

Copyright © 2012 Shaojun Dai and Ruihai Zhang. 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 Merrifield-Simmons index of a graph is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets of The Hosoya index of a graph is defined as the total number of independent edge subsets, that is, the total number of its matchings. By we denote the set of graphs with vertices, cycles, the length of every cycle is , and all the edges not on the cycles are pendant edges which are attached to the same vertex. In this paper, we investigate the Merrifield-Simmons index and the Hosoya index for a graph in .

1. Introduction

Let denote a graph whose set of vertices and set of edges are and , respectively. For any , we denote the neighbors of as , and . By , we denote the number of vertices of . All graphs considered here are both finite and simple. We denote, respectively, by , , and the star, path, and cycle with vertices. For other graph-theoretical terminology and notation, we refer to [1]. By we denote the set of graphs with vertices, cycles, the length of every cycle is and all the edges not on the cycles are pendant edges which are attached to the same vertex, where and the vertex denotes the central vertex of the graphs, as shown in Figure 1. The Merrifield-Simmons index of a graph is defined as the number of subsets of the vertex set, in which any two vertices are nonadjacent, that is, the number of independent vertex sets of . The Hosoya index of a graph is defined as the total number of independent edge subsets, that is, the total number of its matchings. In particular, the Merrifield-Simmons index, and Hosoya index of the empty graph are .

Figure 1: graphs.

The Merrifield-Simmons index was introduced by Merrifield and Simmons [2] in 1989, and the Hosoya index was introduced by Hosoya [3] in 1971. They are one of the topological indices whose mathematical properties turned out to be applicable to several questions of molecular chemistry. For example, the connections with physicochemical properties such as boiling point, entropy or heat of vaporization are well studied.

Several papers deal with the Merrifield-Simmons index and Hosoya index in several given graph classes. Usually, trees, unicyclic graphs, and certain structures involving pentagonal and hexagonal cycles are of major interest [4–12].

In this paper, we investigate the Merrifield-Simmons index and the Hosoya index for a graph in .

2. Some Lemmas

In this section, we gather notations which are used throughout this paper and give some necessary lemmas which will be used to prove our main results.

If and , then and denote the subgraphs of obtained by deleting the edges of and the vertices of , respectively. By denote the smallest positive integer not less than . By we denote the th Fibonacci number, where , with initial conditions and .

The following lemma is obvious.

Lemma 2.1. Let .

(i)If , then . (ii)If , then .

We will make use of the following two well-known lemmas on the Merrifield-Simmons index and Hosoya index.

Lemma 2.2. Let be a graph.

(i)If are the components of the graph , then (see [10, Lemma 1]).(ii)If , then (see [10, Lemma 1]).(iii); for any ; for any (see [13]).

Lemma 2.3. Let be a graph.

(i)If are the components of the graph , then (see [10, Lemma 1]).(ii)If , then (see [14]).(iii)If , then (see [10, Lemma 1]). (iv); for any ; for any (see [14]).

3. The Merrifield-Simmons Index of

In this section, we will give the Merrifield-Simmons index of and their order.

Theorem 3.1. Let , . Then

Proof. By Lemma 2.2 and an elementary calculating, we have

Theorem 3.2. Let , , . Then .

Proof. Let . We have

Obviously, by be exist. Again by , we have

Thus

By Theorem 3.2, we obtain the order of the Merrifield-Simmons index of .

Corollary 3.3. Let , , . Then , and has the largest Merrifield-Simmons index among the graphs in .

Theorem 3.4. Let , , . Then .

Proof. Let , . If , then

If , then

If , then

Obviously, by exists. Again by Lemma 2.1(i), we have

Thus

By Theorem 3.4, we obtain the order of the Merrifield-Simmons index of .

Corollary 3.5. Let , . Then , and has the largest Merrifield-Simmons index of among the graphs in .

4. The Hosoya Index of

In this section, we will give the Hosoya index of and their order.

Theorem 4.1. Let and . Then

Proof. For all and , according to Lemma 2.3, we have the following:

Theorem 4.2. Let , and . Then .

Proof. Let , and . We have

Obviously, by exists. We have

Thus

By Theorem 4.2, we obtain the order of the Hosoya index of .

Corollary 4.3. Let , , . Then , and has the smallest Hosoya index among the graphs in .

Theorem 4.4. Let , , . Then .

Proof. Let , , ,

Obviously, by exists. We have

Thus

By Theorem 4.4, we obtain the order of the Hosoya index of .

Corollary 4.5. Let , , . Then , and has the smallest Hosoya index of among the graphs in .

Acknowledgments

Project 10871205 supported by the National Natural Science Foundation China and the Research Fund of Tianjin Polytechnic University. The authors are very grateful to the anonymous referee for his valuable comments and suggestions, which led to an improvement of the original paper.

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