Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 520156 | 8 pages | https://doi.org/10.1155/2012/520156

The Merrifield-Simmons Index and Hosoya Index of 𝐶(𝑛,𝑘,𝜆) Graphs

Academic Editor: Ferenc Hartung
Received29 May 2012
Revised18 Jun 2012
Accepted27 Jun 2012
Published19 Jul 2012

Abstract

The Merrifield-Simmons index 𝑖(𝐺) of a graph 𝐺 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 𝐺 The Hosoya index 𝑧(𝐺) of a graph 𝐺 is defined as the total number of independent edge subsets, that is, the total number of its matchings. By 𝐶(𝑛,𝑘,𝜆) we denote the set of graphs with 𝑛 vertices, 𝑘 cycles, the length of every cycle is 𝜆, 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 𝑖(𝐺) and the Hosoya index 𝑧(𝐺) for a graph 𝐺 in 𝐶(𝑛,𝑘,𝜆).

1. Introduction

Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) denote a graph whose set of vertices and set of edges are 𝑉(𝐺) and 𝐸(𝐺), respectively. For any 𝑣∈𝑉(𝐺), we denote the neighbors of 𝑣 as 𝑁𝐺(𝑣), and [𝑣]=𝑁𝐺(𝑣)∪{𝑣}. By 𝑛, we denote the number of vertices of 𝐺. All graphs considered here are both finite and simple. We denote, respectively, by 𝑆𝑛, 𝑃𝑛, and 𝐶𝑛 the star, path, and cycle with 𝑛 vertices. For other graph-theoretical terminology and notation, we refer to [1]. By 𝐶(𝑛,𝑘,𝜆) we denote the set of graphs with 𝑛 vertices, 𝑘 cycles, the length of every cycle is 𝜆 and all the edges not on the cycles are pendant edges which are attached to the same vertex, where 𝑛1=𝑛−[(𝜆−1)𝑘+1]≥0 and the vertex 𝑣 denotes the central vertex of the graphs, as shown in Figure 1. The Merrifield-Simmons index 𝑖(𝐺) of a graph 𝐺 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 𝐺. The Hosoya index 𝑧(𝐺) of a graph 𝐺 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 1.

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 𝑖(𝐺) and the Hosoya index 𝑧(𝐺) for a graph 𝐺 in 𝐶(𝑛,𝑘,𝜆).

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 𝐸′⊆𝐸(𝐺) and 𝑊⊆𝑉(𝐺), then 𝐺−𝐸′ and 𝐺−𝑊 denote the subgraphs of 𝐺 obtained by deleting the edges of 𝐸′ and the vertices of 𝑊, respectively. By ⌊𝑥⌋ denote the smallest positive integer not less than 𝑥. By 𝑓(𝑛) we denote the 𝑛th Fibonacci number, where 𝑛∈ℕ, 𝑓(𝑛)+𝑓(𝑛+1)=𝑓(𝑛+2) with initial conditions 𝑓(0)=0 and 𝑓(1)=1.

The following lemma is obvious.

Lemma 2.1. Let 𝑛∈𝑁. (i)If 𝑛≥6, then 𝑓(𝑛)≤2𝑛−3. (ii)If 𝑛≥2, then 𝑓(𝑛)≥𝑛/2.

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

Lemma 2.2. Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a graph. (i)If 𝐺1,𝐺2,…,𝐺𝑚 are the components of the graph 𝐺, then ∏𝑖(𝐺)=𝑚𝑖=1𝑖(𝐺𝑖) (see [10, Lemma 1]).(ii)If 𝑥∈𝑉(𝐺), then 𝑖(𝐺)=𝑖(𝐺−{𝑥})+𝑖(𝐺−[𝑥]) (see [10, Lemma 1]).(iii)𝑖(𝑆𝑛)=2𝑛−1+1; 𝑖(𝑃𝑛)=𝑓(𝑛+2) for any 𝑛∈ℕ; 𝑖(𝐶𝑛)=𝑓(𝑛−1)+𝑓(𝑛+1) for any 𝑛≥3 (see [13]).

Lemma 2.3. Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a graph. (i)If 𝐺1,𝐺2,…,𝐺𝑚 are the components of the graph 𝐺, then ∏𝑧(𝐺)=𝑚𝑖=1𝑧(𝐺𝑖) (see [10, Lemma 1]).(ii)If 𝑒=𝑥𝑦∈𝐸(𝐺), then 𝑧(𝐺)=𝑧(𝐺−{𝑒})+𝑧(𝐺−{𝑥,𝑦}) (see [14]).(iii)If 𝑥∈𝑉(𝐺), then ∑𝑧(𝐺)=𝑧(𝐺−{𝑥})+𝑦∈𝑁𝐺(𝑥)𝑧(𝐺−{𝑥,𝑦}) (see [10, Lemma 1]). (iv)𝑧(𝑆𝑛)=𝑛; 𝑧(𝑃𝑛)=𝑓(𝑛+1) for any 𝑛∈ℕ; 𝑧(𝐶𝑛)=𝑓(𝑛−1)+𝑓(𝑛+1) for any 𝑛≥3 (see [14]).

3. The Merrifield-Simmons Index of 𝐶(𝑛,𝑘,𝜆)

In this section, we will give the Merrifield-Simmons index of 𝐶(𝑛,𝑘,𝜆) and their order.

Theorem 3.1. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆−1)⌋, 𝜆≥3. Then 𝑖(𝐶(𝑛,𝑘,𝜆))=2[𝑛−(𝜆−1)𝑘−1]𝑓(𝜆+1)𝑘+𝑓(𝜆−1)𝑘.(3.1)

Proof. By Lemma 2.2 and an elementary calculating, we have [𝑣]𝑃𝑖(𝐶(𝑛,𝑘,𝜆))=𝑖(𝐺−{𝑣})+𝑖(𝐺−)=𝑖1[𝑛−(𝜆−1)𝑘−1]𝑖(𝑃𝜆−1)𝑘+𝑖𝑃𝜆−3𝑘=2[𝑛−(𝜆−1)𝑘−1]𝑓(𝜆+1)𝑘+𝑓(𝜆−1)𝑘.(3.2)

Theorem 3.2. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1, 𝑘0≥1, 𝑛≥5. Then 𝑖(𝐶(𝑛,𝑘0,𝜆+1))<𝑖(𝐶(𝑛,𝑘0,𝜆)).

Proof. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1. We have Δ1𝐶=𝑖𝑛,𝑘0𝐶,𝜆+1−𝑖𝑛,𝑘0,𝜆=2(𝑛−𝜆𝑘0−1)𝑓(𝜆+2)𝑘0+𝑓(𝜆)𝑘0−2[𝑛−𝑘0(𝜆−1)−1]𝑓(𝜆+1)𝑘0−𝑓(𝜆−1)𝑘0=2(𝑛−𝜆𝑘0−1)𝑓(𝜆+2)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0.(3.3)
Obviously, 2(𝑛−𝜆𝑘0−1)≥1 by 𝑧(𝐶(𝑛,𝑘0,𝜆+1)) be exist. Again by 𝑓(𝜆+2)<2𝑓(𝜆+1), we have 2(𝑛−𝜆𝑘0−1)𝑓(𝜆+2)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0≤𝑓(𝜆+2)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0.(3.4) Thus Δ1≤𝑓(𝜆+2)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0=[]𝑓(𝜆+1)+𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0=𝑓(𝜆+1)𝑘0+𝑘0𝑓(𝜆+1)𝑘0−1𝑓(𝜆)+⋯+𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0<2𝑘0−1𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆+1)𝑘0+𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0=−𝑓(𝜆+1)𝑘0+2𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0[]=−𝑓(𝜆)+𝑓(𝜆−1)𝑘0+2𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0<−𝑓(𝜆)𝑘0+2𝑘0−1𝑓(𝜆−1)𝑘0+2𝑓(𝜆)𝑘0−𝑓(𝜆−1)𝑘0=−2𝑘0𝑓(𝜆−1)𝑘0+𝑓(𝜆)𝑘0(by𝑓(𝜆)<2𝑓(𝜆−1))=−(2𝑓(𝜆−1))𝑘0−𝑓(𝜆)𝑘0<0.(3.5)
By Theorem 3.2, we obtain the order of the Merrifield-Simmons index of 𝐶(𝑛,𝑘0,𝜆).

Corollary 3.3. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1, 𝑘0≥1, 𝑛≥5. Then 𝑖(𝐶(𝑛,𝑘0,3))>𝑖(𝐶(𝑛,𝑘0,4))>𝑖(𝐶(𝑛,𝑘0,5))>⋯, and 𝐶(𝑛,𝑘0,3) has the largest Merrifield-Simmons index among the graphs in 𝐶(𝑛,𝑘0,𝜆).

Theorem 3.4. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆0−1)⌋, 𝜆0≥3, 𝑛≥3. Then 𝑖(𝐶(𝑛,𝑘+1,𝜆0))<𝑖(𝐶(𝑛,𝑘,𝜆0)).

Proof. Let 𝑘≥1, 𝑛≥3. If 𝜆0=3, then Δ2=𝑖(𝐶(𝑛,𝑘+1,3))−𝑖(𝐶(𝑛,𝑘,3))=2[𝑛−2(𝑘+1)−1]𝑓(4)𝑘+1+𝑓(2)𝑘+1−2[𝑛−2𝑘−1]𝑓(4)𝑘−𝑓(2)𝑘=2𝑛−134𝑘+1−2𝑛−134𝑘=−2𝑛−134𝑘⋅14<0.(3.6) If 𝜆0=4, then Δ3=𝑖(𝐶(𝑛,𝑘+1,4))−𝑖(𝐶(𝑛,𝑘,4))=2[𝑛−3(𝑘+1)−1]𝑓(5)𝑘+1+𝑓(3)𝑘+1−2[𝑛−3𝑘−1]𝑓(5)𝑘−𝑓(3)𝑘=2𝑛−3𝑘−45𝑘(−3)+2𝑘≤−3⋅5𝑘+2𝑘<0.(3.7) If 𝜆0≥5, then Δ4𝐶=𝑖𝑛,𝑘+1,𝜆0𝐶−𝑖𝑛,𝑘,𝜆0=2[𝑛−(𝜆0−1)(𝑘+1)−1]𝑓𝜆0+1𝑘+1𝜆+𝑓0−1𝑘+1−2[𝑛−(𝜆0−1)𝑘−1]𝑓𝜆0+1𝑘𝜆−𝑓0−1𝑘=2[𝑛−(𝜆0−1)(𝑘+1)−1]𝑓𝜆0+1𝑘𝑓𝜆0+1−2𝜆0−1𝜆+𝑓0−1𝑘+1𝜆−𝑓0−1𝑘.(3.8) Obviously, 2[𝑛−(𝜆0−1)(𝑘+1)−1]≥1 by 𝑧(𝐶(𝑛,𝑘+1,𝜆0)) exists. Again by Lemma 2.1(i), we have 𝑓𝜆0+1−2𝜆0−1𝜆≤−𝑓0+1≤0.(3.9) Thus Δ4𝜆≤𝑓0+1𝑘𝑓𝜆0+1−2𝜆0−1𝜆+𝑓0−1𝑘+1𝜆−𝑓0−1𝑘𝜆≤𝑓0+1𝑘𝜆−𝑓0𝜆+1+𝑓0−1𝑘+1𝜆−𝑓0−1𝑘𝑓𝜆=−0+1𝑘+1𝜆−𝑓0−1𝑘+1𝜆−𝑓0−1𝑘<0.(3.10)
By Theorem 3.4, we obtain the order of the Merrifield-Simmons index of 𝐶(𝑛,𝑘,𝜆0).

Corollary 3.5. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆0−1)⌋, 𝑛≥3. Then 𝑖(𝐶(𝑛,1,𝜆0))>𝑖(𝐶(𝑛,2,𝜆0))>𝑖(𝐶(𝑛,3,𝜆0))>⋯, and 𝐶(𝑛,1,𝜆0) has the largest Merrifield-Simmons index of among the graphs in 𝐶(𝑛,𝑘,𝜆0).

4. The Hosoya Index of 𝐶(𝑛,𝑘,𝜆)

In this section, we will give the Hosoya index of 𝐶(𝑛,𝑘,𝜆) and their order.

Theorem 4.1. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆−1)⌋ and 𝜆≥3. Then []𝑧(𝐶(𝑛,𝑘,𝜆))=𝑛−(𝜆−1)𝑘𝑓(𝜆)𝑘+2𝑘𝑓(𝜆−1)𝑓(𝜆)𝑘−1.(4.1)

Proof. For all 1≤𝑘≤⌊(𝑛−1)/(𝜆−1)⌋ and 𝜆≥3, according to Lemma 2.3, we have the following: 𝑧(𝐶(𝑛,𝑘,𝜆))=𝑧(𝐺−{𝑣})+𝑥∈𝑁𝐺(𝑣)=𝑧(𝐺−{𝑥,𝑣})𝑧(𝑃𝜆−1)𝑘+[]𝑛−(𝜆−1)𝑘−1𝑧(𝑃𝜆−1)𝑘+2𝑘𝑧(𝑃𝜆−2),𝑧(𝑃𝜆−1)𝑘−1=𝑓(𝜆)𝑘+[]𝑓𝑛−(𝜆−1)𝑘−1(𝜆)𝑘+2𝑘𝑓(𝜆−1)𝑓(𝜆)𝑘−1=[]𝑓𝑛−(𝜆−1)𝑘(𝜆)𝑘+2𝑘𝑓(𝜆−1)𝑓(𝜆)𝑘−1.(4.2)

Theorem 4.2. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1, 𝑘0≥1 and 𝑛≥5. Then 𝑧(𝐶(𝑛,𝑘0,𝜆))<𝑧(𝐶(𝑛,𝑘0,𝜆+1)).

Proof. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1, 𝑘0≥1 and 𝑛≥5. We have Δ5𝐶=𝑧𝑛,𝑘0𝐶,𝜆+1−𝑧𝑛,𝑘0=,𝜆𝑛−𝜆𝑘0𝑓(𝜆+1)𝑘0+2𝑘0𝑓(𝜆)𝑓(𝜆+1)𝑘0−1−𝑛−𝜆𝑘0+𝑘0𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆−1)𝑓(𝜆)𝑘0−1=𝑛−𝜆𝑘0𝑓(𝜆+1)𝑘0−𝑓(𝜆)𝑘0+2𝑘0𝑓(𝜆)𝑓(𝜆+1)𝑘0−1−𝑘0𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆−1)𝑓(𝜆)𝑘0−1.(4.3)
Obviously, (𝑛−𝜆𝑘0)≥1 by 𝑧(𝐶(𝑛,𝑘0,𝜆+1)) exists. We have 𝑛−𝜆𝑘0𝑓(𝜆+1)𝑘0−𝑓(𝜆)𝑘0≥𝑓(𝜆+1)𝑘0−𝑓(𝜆)𝑘0=[𝑓](𝜆)+𝑓(𝜆−1)𝑘0−1−𝑓(𝜆)𝑘0≥𝑓(𝜆)𝑘0+𝑘0𝑓(𝜆)𝑘0−1𝑓(𝜆−1)−𝑓(𝜆)𝑘0=𝑘0𝑓(𝜆)𝑘0−1𝑓(𝜆−1),2𝑘0𝑓(𝜆)𝑓(𝜆+1)𝑘0−1=2𝑘0𝑓[](𝜆)𝑓(𝜆)+𝑓(𝜆−1)𝑘0−1≥2𝑘0𝑓𝑓(𝜆)(𝜆)𝑘0−1+𝑘0𝑓−1(𝜆)𝑘0−2𝑓(𝜆−1)=2𝑘0𝑓(𝜆)𝑘0+2𝑘0𝑘0−1𝑓(𝜆)𝑘0−1𝑓(𝜆−1).(4.4)
Thus Δ5≥𝑘0𝑓(𝜆)𝑘0−1𝑓(𝜆−1)+2𝑘0𝑓(𝜆)𝑘0+2𝑘0𝑘0−1𝑓(𝜆)𝑘0−1𝑓(𝜆−1)−𝑘0𝑓(𝜆)𝑘0−2𝑘0𝑓(𝜆−1)𝑓(𝜆)𝑘0−1=𝑘0𝑓(𝜆)𝑘0+2𝑘0𝑘0−32𝑓(𝜆)𝑘0−1𝑓(𝜆−1)>0.(4.5)
By Theorem 4.2, we obtain the order of the Hosoya index of 𝐶(𝑛,𝑘0,𝜆).

Corollary 4.3. Let 3≤𝜆≤⌊(𝑛−1)/𝑘0⌋+1, 𝑘0≥1, 𝑛≥5. Then 𝑧(𝐶(𝑛,𝑘0,3))<𝑧(𝐶(𝑛,𝑘0,4))<𝑧(𝐶(𝑛,𝑘0,5))<⋯, and 𝐶(𝑛,𝑘0,3) has the smallest Hosoya index among the graphs in 𝐶(𝑛,𝑘0,𝜆).

Theorem 4.4. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆0−1)⌋, 𝜆0≥3, 𝑛≥3. Then 𝑧(𝐶(𝑛,𝑘,𝜆0))<𝑧(𝐶(𝑛,𝑘+1,𝜆0)).

Proof. Let 𝑘≥1, 𝜆0≥3, 𝑛≥3, Δ6𝐶=𝑧𝑛,𝑘+1,𝜆0𝐶−𝑧𝑛,𝑘,𝜆0=𝜆𝑛−0𝑓𝜆−1(𝑘+1)0𝑘+1𝜆+2(𝑘+1)𝑓0𝑓𝜆−10𝑘−𝜆𝑛−0𝑘𝑓𝜆−10𝑘𝜆−2𝑘𝑓0𝑓𝜆−10𝑘−1=𝜆𝑛−0𝜆−1(𝑘+1),𝑓0𝑘+1𝜆−𝑓0𝑘−𝜆0𝑓𝜆−10𝑘𝜆+2𝑘𝑓0𝑓𝜆−10𝑘𝜆−𝑓0𝑘−1𝜆+2𝑓0𝑓𝜆−10𝑘.(4.6)
Obviously, 𝑛−(𝜆0−1)(𝑘+1)≥1 by 𝑧(𝐶(𝑛,𝑘+1,𝜆0)) exists. We have 𝜆𝑛−0𝑓𝜆−1(𝑘+1)0𝑘+1𝜆−𝑓0𝑘𝜆≥0,2𝑘𝑓0𝑓𝜆−10𝑘𝜆−𝑓0𝑘−1>0.(4.7)
Thus Δ6𝜆>2𝑓0𝑓𝜆−10𝑘−𝜆0𝑓𝜆−10𝑘𝜆=𝑓0𝑘𝜆2𝑓0−𝜆−10−1≥0(byLemma2.1(ii)).(4.8)
By Theorem 4.4, we obtain the order of the Hosoya index of 𝐶(𝑛,𝑘,𝜆0).

Corollary 4.5. Let 1≤𝑘≤⌊(𝑛−1)/(𝜆0−1)⌋, 𝜆0≥3, 𝑛≥3. Then 𝑧(𝐶(𝑛,1,𝜆0))<𝑧(𝐶(𝑛,2,𝜆0))<𝑧(𝐶(𝑛,2,𝜆0))<⋯, and 𝐶(𝑛,1,𝜆0) has the smallest Hosoya index of among the graphs in 𝐶(𝑛,𝑘,𝜆0).

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.

References

  1. R. Diestel, Graph Theory, vol. 173 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 2000. View at: Publisher Site | Zentralblatt MATH
  2. R. E. Merrifield and H. E. Simmons, Topological Methods in Chemistry, John Wiley & Sons, New York, NY, USA, 1989.
  3. H. Hosoya, “Topological index, a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons,” Bulletin of the Chemical Society of Japan, vol. 44, pp. 2322–2339, 1971. View at: Google Scholar
  4. H. Deng and S. Chen, “The extremal unicyclic graphs with respect to Hosoya index and Merrifield-Simmons index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 59, no. 1, pp. 171–190, 2008. View at: Google Scholar | Zentralblatt MATH
  5. M. Fischermann, I. Gutman, A. Hoffmann, D. Rautenbach, D. Vidovic, and L. and Volkmann, “Extremal chemical trees,” Zeitschrift für Naturforschung Section B-A Journal of Chemical Sciences, vol. 57, pp. 49–52, 2002. View at: Google Scholar
  6. M. Fischermann, L. Volkmann, and D. Rautenbach, “A note on the number of matchings and independent sets in trees,” Discrete Applied Mathematics, vol. 145, no. 3, pp. 483–489, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. I. Gutman, “Extremal hexagonal chains,” Journal of Mathematical Chemistry, vol. 12, no. 1–4, pp. 197–210, 1993. View at: Publisher Site | Google Scholar
  8. I. Gutman and F. J. Zhang, “On the ordering of graphs with respect to their matching numbers,” Discrete Applied Mathematics, vol. 15, no. 1, pp. 25–33, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. Y. Hou, “On acyclic systems with minimal Hosoya index,” Discrete Applied Mathematics, vol. 119, no. 3, pp. 251–257, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. S. G. Wagner, “Extremal trees with respect to Hosoya index and Merrifield-Simmons index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 57, no. 1, pp. 221–233, 2007. View at: Google Scholar | Zentralblatt MATH
  11. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, Germany, 1986. View at: Publisher Site
  12. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, NY, USA, 1976.
  13. H. Deng, “The smallest Merrifield-Simmons index of (n,n+1)-graphs,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 320–326, 2009. View at: Publisher Site | (n,n+1)-graphs&author=H. Deng&publication_year=2009" target="_blank">Google Scholar
  14. H. Deng, “The largest Hosoya index of (n,n+1)-graphs,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2499–2506, 2008. View at: Publisher Site | (n,n+1)-graphs&author=H. Deng&publication_year=2008" target="_blank">Google Scholar

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.

1311 Views | 406 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder