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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 520156, 8 pages
http://dx.doi.org/10.1155/2012/520156
Research Article

The Merrifield-Simmons Index and Hosoya Index of 𝐢(𝑛,π‘˜,πœ†) Graphs

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: FerencΒ Hartung

Copyright Β© 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 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 𝐢(𝑛,π‘˜,πœ†) we denote the set of graphs with 𝑛 vertices, π‘˜ cycles, the length of every cycle is πœ†, 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 𝐢(𝑛,π‘˜,πœ†).

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 π‘£βˆˆπ‘‰(𝐺), we denote the neighbors of 𝑣 as 𝑁𝐺(𝑣), and [𝑣]=𝑁𝐺(𝑣)βˆͺ{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 𝐢𝑛 the star, path, and cycle with 𝑛 vertices. For other graph-theoretical terminology and notation, we refer to [1]. By 𝐢(𝑛,π‘˜,πœ†) we denote the set of graphs with 𝑛 vertices, π‘˜ cycles, the length of every cycle is πœ† and all the edges not on the cycles are pendant edges which are attached to the same vertex, where 𝑛1=π‘›βˆ’[(πœ†βˆ’1)π‘˜+1]β‰₯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.

520156.fig.001
Figure 1: 𝐢(𝑛,π‘˜,πœ†) 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 𝑖(𝐺) and the Hosoya index 𝑧(𝐺) for a graph 𝐺 in 𝐢(𝑛,π‘˜,πœ†).

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 πΈβ€²βŠ†πΈ(𝐺) and π‘ŠβŠ†π‘‰(𝐺), then πΊβˆ’πΈβ€² and πΊβˆ’π‘Š denote the subgraphs of 𝐺 obtained by deleting the edges of 𝐸′ and the vertices of π‘Š, respectively. By ⌊π‘₯βŒ‹ denote the smallest positive integer not less than π‘₯. By 𝑓(𝑛) we denote the 𝑛th Fibonacci number, where π‘›βˆˆβ„•, 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 π‘›βˆˆπ‘. (i)If 𝑛β‰₯6, then 𝑓(𝑛)≀2π‘›βˆ’3. (ii)If 𝑛β‰₯2, then 𝑓(𝑛)β‰₯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,…,πΊπ‘š are the components of the graph 𝐺, then βˆπ‘–(𝐺)=π‘šπ‘–=1𝑖(𝐺𝑖) (see [10, Lemma 1]).(ii)If π‘₯βˆˆπ‘‰(𝐺), then 𝑖(𝐺)=𝑖(πΊβˆ’{π‘₯})+𝑖(πΊβˆ’[π‘₯]) (see [10, Lemma 1]).(iii)𝑖(𝑆𝑛)=2π‘›βˆ’1+1; 𝑖(𝑃𝑛)=𝑓(𝑛+2) for any π‘›βˆˆβ„•; 𝑖(𝐢𝑛)=𝑓(π‘›βˆ’1)+𝑓(𝑛+1) for any 𝑛β‰₯3 (see [13]).

Lemma 2.3. Let 𝐺=(𝑉(𝐺),𝐸(𝐺)) be a graph. (i)If 𝐺1,𝐺2,…,πΊπ‘š are the components of the graph 𝐺, then βˆπ‘§(𝐺)=π‘šπ‘–=1𝑧(𝐺𝑖) (see [10, Lemma 1]).(ii)If 𝑒=π‘₯π‘¦βˆˆπΈ(𝐺), then 𝑧(𝐺)=𝑧(πΊβˆ’{𝑒})+𝑧(πΊβˆ’{π‘₯,𝑦}) (see [14]).(iii)If π‘₯βˆˆπ‘‰(𝐺), then βˆ‘π‘§(𝐺)=𝑧(πΊβˆ’{π‘₯})+π‘¦βˆˆπ‘πΊ(π‘₯)𝑧(πΊβˆ’{π‘₯,𝑦}) (see [10, Lemma 1]). (iv)𝑧(𝑆𝑛)=𝑛; 𝑧(𝑃𝑛)=𝑓(𝑛+1) for any π‘›βˆˆβ„•; 𝑧(𝐢𝑛)=𝑓(π‘›βˆ’1)+𝑓(𝑛+1) for any 𝑛β‰₯3 (see [14]).

3. The Merrifield-Simmons Index of 𝐢(𝑛,π‘˜,πœ†)

In this section, we will give the Merrifield-Simmons index of 𝐢(𝑛,π‘˜,πœ†) and their order.

Theorem 3.1. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†βˆ’1)βŒ‹, πœ†β‰₯3. Then 𝑖(𝐢(𝑛,π‘˜,πœ†))=2[π‘›βˆ’(πœ†βˆ’1)π‘˜βˆ’1]𝑓(πœ†+1)π‘˜+𝑓(πœ†βˆ’1)π‘˜.(3.1)

Proof. By Lemma 2.2 and an elementary calculating, we have [𝑣]𝑃𝑖(𝐢(𝑛,π‘˜,πœ†))=𝑖(πΊβˆ’{𝑣})+𝑖(πΊβˆ’)=𝑖1ξ€Έ[π‘›βˆ’(πœ†βˆ’1)π‘˜βˆ’1]𝑖(π‘ƒπœ†βˆ’1)ξ€»π‘˜+ξ€Ίπ‘–ξ€·π‘ƒπœ†βˆ’3ξ€Έξ€»π‘˜=2[π‘›βˆ’(πœ†βˆ’1)π‘˜βˆ’1]𝑓(πœ†+1)π‘˜+𝑓(πœ†βˆ’1)π‘˜.(3.2)

Theorem 3.2. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1, π‘˜0β‰₯1, 𝑛β‰₯5. Then 𝑖(𝐢(𝑛,π‘˜0,πœ†+1))<𝑖(𝐢(𝑛,π‘˜0,πœ†)).

Proof. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1. We have Ξ”1𝐢=𝑖𝑛,π‘˜0𝐢,πœ†+1ξ€Έξ€Έβˆ’π‘–π‘›,π‘˜0,πœ†ξ€Έξ€Έ=2(π‘›βˆ’πœ†π‘˜0βˆ’1)𝑓(πœ†+2)π‘˜0+𝑓(πœ†)π‘˜0βˆ’2[π‘›βˆ’π‘˜0(πœ†βˆ’1)βˆ’1]𝑓(πœ†+1)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0=2(π‘›βˆ’πœ†π‘˜0βˆ’1)𝑓(πœ†+2)π‘˜0βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0ξ€»+𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0ξ€».(3.3)
Obviously, 2(π‘›βˆ’πœ†π‘˜0βˆ’1)β‰₯1 by 𝑧(𝐢(𝑛,π‘˜0,πœ†+1)) be exist. Again by 𝑓(πœ†+2)<2𝑓(πœ†+1), we have 2(π‘›βˆ’πœ†π‘˜0βˆ’1)𝑓(πœ†+2)π‘˜0βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0≀𝑓(πœ†+2)π‘˜0βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0.(3.4) Thus Ξ”1≀𝑓(πœ†+2)π‘˜0βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0+𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0=[]𝑓(πœ†+1)+𝑓(πœ†)π‘˜0βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0+𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0ξ€»=𝑓(πœ†+1)π‘˜0+π‘˜0𝑓(πœ†+1)π‘˜0βˆ’1𝑓(πœ†)+β‹―+𝑓(πœ†)π‘˜0ξ€»βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0+𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0ξ€»<2ξ€Ίξ€·π‘˜0ξ€Έβˆ’1𝑓(πœ†+1)π‘˜0+𝑓(πœ†)π‘˜0ξ€»βˆ’2π‘˜0𝑓(πœ†+1)π‘˜0+𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0ξ€»=βˆ’π‘“(πœ†+1)π‘˜0+2𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0[]=βˆ’π‘“(πœ†)+𝑓(πœ†βˆ’1)π‘˜0+2𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0ξ€Ί<βˆ’π‘“(πœ†)π‘˜0+ξ€·2π‘˜0ξ€Έβˆ’1𝑓(πœ†βˆ’1)π‘˜0ξ€»+2𝑓(πœ†)π‘˜0βˆ’π‘“(πœ†βˆ’1)π‘˜0=βˆ’2π‘˜0𝑓(πœ†βˆ’1)π‘˜0+𝑓(πœ†)π‘˜0ξ€Ί(by𝑓(πœ†)<2𝑓(πœ†βˆ’1))=βˆ’(2𝑓(πœ†βˆ’1))π‘˜0βˆ’π‘“(πœ†)π‘˜0ξ€»<0.(3.5)
By Theorem 3.2, we obtain the order of the Merrifield-Simmons index of 𝐢(𝑛,π‘˜0,πœ†).

Corollary 3.3. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1, π‘˜0β‰₯1, 𝑛β‰₯5. Then 𝑖(𝐢(𝑛,π‘˜0,3))>𝑖(𝐢(𝑛,π‘˜0,4))>𝑖(𝐢(𝑛,π‘˜0,5))>β‹―, and 𝐢(𝑛,π‘˜0,3) has the largest Merrifield-Simmons index among the graphs in 𝐢(𝑛,π‘˜0,πœ†).

Theorem 3.4. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†0βˆ’1)βŒ‹, πœ†0β‰₯3, 𝑛β‰₯3. Then 𝑖(𝐢(𝑛,π‘˜+1,πœ†0))<𝑖(𝐢(𝑛,π‘˜,πœ†0)).

Proof. Let π‘˜β‰₯1, 𝑛β‰₯3. If πœ†0=3, then Ξ”2=𝑖(𝐢(𝑛,π‘˜+1,3))βˆ’π‘–(𝐢(𝑛,π‘˜,3))=2[π‘›βˆ’2(π‘˜+1)βˆ’1]𝑓(4)π‘˜+1+𝑓(2)π‘˜+1βˆ’2[π‘›βˆ’2π‘˜βˆ’1]𝑓(4)π‘˜βˆ’π‘“(2)π‘˜=2π‘›βˆ’1ξ‚€34ξ‚π‘˜+1βˆ’2π‘›βˆ’1ξ‚€34ξ‚π‘˜=βˆ’2π‘›βˆ’1ξ‚€34ξ‚π‘˜β‹…14<0.(3.6) If πœ†0=4, then Ξ”3=𝑖(𝐢(𝑛,π‘˜+1,4))βˆ’π‘–(𝐢(𝑛,π‘˜,4))=2[π‘›βˆ’3(π‘˜+1)βˆ’1]𝑓(5)π‘˜+1+𝑓(3)π‘˜+1βˆ’2[π‘›βˆ’3π‘˜βˆ’1]𝑓(5)π‘˜βˆ’π‘“(3)π‘˜=2π‘›βˆ’3π‘˜βˆ’45π‘˜(βˆ’3)+2π‘˜β‰€βˆ’3β‹…5π‘˜+2π‘˜<0.(3.7) If πœ†0β‰₯5, then Ξ”4𝐢=𝑖𝑛,π‘˜+1,πœ†0ξ€·πΆξ€·ξ€Έξ€Έβˆ’π‘–π‘›,π‘˜,πœ†0ξ€Έξ€Έ=2[π‘›βˆ’(πœ†0βˆ’1)(π‘˜+1)βˆ’1]π‘“ξ€·πœ†0ξ€Έ+1π‘˜+1ξ€·πœ†+𝑓0ξ€Έβˆ’1π‘˜+1βˆ’2[π‘›βˆ’(πœ†0βˆ’1)π‘˜βˆ’1]π‘“ξ€·πœ†0ξ€Έ+1π‘˜ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜=2[π‘›βˆ’(πœ†0βˆ’1)(π‘˜+1)βˆ’1]π‘“ξ€·πœ†0ξ€Έ+1π‘˜ξ€Ίπ‘“ξ€·πœ†0ξ€Έ+1βˆ’2πœ†0βˆ’1ξ€»ξ€·πœ†+𝑓0ξ€Έβˆ’1π‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜.(3.8) Obviously, 2[π‘›βˆ’(πœ†0βˆ’1)(π‘˜+1)βˆ’1]β‰₯1 by 𝑧(𝐢(𝑛,π‘˜+1,πœ†0)) exists. Again by Lemma 2.1(i), we have π‘“ξ€·πœ†0ξ€Έ+1βˆ’2πœ†0βˆ’1ξ€·πœ†β‰€βˆ’π‘“0ξ€Έ+1≀0.(3.9) Thus Ξ”4ξ€·πœ†β‰€π‘“0ξ€Έ+1π‘˜ξ€Ίπ‘“ξ€·πœ†0ξ€Έ+1βˆ’2πœ†0βˆ’1ξ€»ξ€·πœ†+𝑓0ξ€Έβˆ’1π‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜ξ€·πœ†β‰€π‘“0ξ€Έ+1π‘˜ξ€Ίξ€·πœ†βˆ’π‘“0ξ€·πœ†+1ξ€Έξ€»+𝑓0ξ€Έβˆ’1π‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜ξ‚ƒπ‘“ξ€·πœ†=βˆ’0ξ€Έ+1π‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜+1ξ‚„ξ€·πœ†βˆ’π‘“0ξ€Έβˆ’1π‘˜<0.(3.10)
By Theorem 3.4, we obtain the order of the Merrifield-Simmons index of 𝐢(𝑛,π‘˜,πœ†0).

Corollary 3.5. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†0βˆ’1)βŒ‹, 𝑛β‰₯3. Then 𝑖(𝐢(𝑛,1,πœ†0))>𝑖(𝐢(𝑛,2,πœ†0))>𝑖(𝐢(𝑛,3,πœ†0))>β‹―, and 𝐢(𝑛,1,πœ†0) has the largest Merrifield-Simmons index of among the graphs in 𝐢(𝑛,π‘˜,πœ†0).

4. The Hosoya Index of 𝐢(𝑛,π‘˜,πœ†)

In this section, we will give the Hosoya index of 𝐢(𝑛,π‘˜,πœ†) and their order.

Theorem 4.1. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†βˆ’1)βŒ‹ and πœ†β‰₯3. Then []𝑧(𝐢(𝑛,π‘˜,πœ†))=π‘›βˆ’(πœ†βˆ’1)π‘˜π‘“(πœ†)π‘˜+2π‘˜π‘“(πœ†βˆ’1)𝑓(πœ†)π‘˜βˆ’1.(4.1)

Proof. For all 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†βˆ’1)βŒ‹ and πœ†β‰₯3, according to Lemma 2.3, we have the following: 𝑧(𝐢(𝑛,π‘˜,πœ†))=𝑧(πΊβˆ’{𝑣})+π‘₯βˆˆπ‘πΊ(𝑣)=𝑧(πΊβˆ’{π‘₯,𝑣})𝑧(π‘ƒπœ†βˆ’1)ξ€»π‘˜+[]ξ€Ίπ‘›βˆ’(πœ†βˆ’1)π‘˜βˆ’1𝑧(π‘ƒπœ†βˆ’1)ξ€»π‘˜ξ€Ί+2π‘˜π‘§(π‘ƒπœ†βˆ’2),𝑧(π‘ƒπœ†βˆ’1)ξ€»π‘˜βˆ’1=𝑓(πœ†)π‘˜+[]π‘“π‘›βˆ’(πœ†βˆ’1)π‘˜βˆ’1(πœ†)π‘˜+2π‘˜π‘“(πœ†βˆ’1)𝑓(πœ†)π‘˜βˆ’1=[]π‘“π‘›βˆ’(πœ†βˆ’1)π‘˜(πœ†)π‘˜+2π‘˜π‘“(πœ†βˆ’1)𝑓(πœ†)π‘˜βˆ’1.(4.2)

Theorem 4.2. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1, π‘˜0β‰₯1 and 𝑛β‰₯5. Then 𝑧(𝐢(𝑛,π‘˜0,πœ†))<𝑧(𝐢(𝑛,π‘˜0,πœ†+1)).

Proof. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1, π‘˜0β‰₯1 and 𝑛β‰₯5. We have Ξ”5𝐢=𝑧𝑛,π‘˜0𝐢,πœ†+1ξ€Έξ€Έβˆ’π‘§π‘›,π‘˜0=ξ€·,πœ†ξ€Έξ€Έπ‘›βˆ’πœ†π‘˜0𝑓(πœ†+1)π‘˜0+2π‘˜0𝑓(πœ†)𝑓(πœ†+1)π‘˜0βˆ’1βˆ’ξ€·π‘›βˆ’πœ†π‘˜0+π‘˜0𝑓(πœ†)π‘˜0βˆ’2π‘˜0𝑓(πœ†βˆ’1)𝑓(πœ†)π‘˜0βˆ’1=ξ€·π‘›βˆ’πœ†π‘˜0𝑓(πœ†+1)π‘˜0βˆ’π‘“(πœ†)π‘˜0ξ€»+2π‘˜0𝑓(πœ†)𝑓(πœ†+1)π‘˜0βˆ’1βˆ’π‘˜0𝑓(πœ†)π‘˜0βˆ’2π‘˜0𝑓(πœ†βˆ’1)𝑓(πœ†)π‘˜0βˆ’1.(4.3)
Obviously, (π‘›βˆ’πœ†π‘˜0)β‰₯1 by 𝑧(𝐢(𝑛,π‘˜0,πœ†+1)) exists. We have ξ€·π‘›βˆ’πœ†π‘˜0𝑓(πœ†+1)π‘˜0βˆ’π‘“(πœ†)π‘˜0ξ€»β‰₯𝑓(πœ†+1)π‘˜0βˆ’π‘“(πœ†)π‘˜0=[𝑓](πœ†)+𝑓(πœ†βˆ’1)π‘˜0βˆ’1βˆ’π‘“(πœ†)π‘˜0β‰₯𝑓(πœ†)π‘˜0+π‘˜0𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1)βˆ’π‘“(πœ†)π‘˜0=π‘˜0𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1),2π‘˜0𝑓(πœ†)𝑓(πœ†+1)π‘˜0βˆ’1=2π‘˜0𝑓[](πœ†)𝑓(πœ†)+𝑓(πœ†βˆ’1)π‘˜0βˆ’1β‰₯2π‘˜0𝑓𝑓(πœ†)(πœ†)π‘˜0βˆ’1+ξ€·π‘˜0ξ€Έπ‘“βˆ’1(πœ†)π‘˜0βˆ’2𝑓(πœ†βˆ’1)=2π‘˜0𝑓(πœ†)π‘˜0+2π‘˜0ξ€·π‘˜0ξ€Έβˆ’1𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1).(4.4)
Thus Ξ”5β‰₯π‘˜0𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1)+2π‘˜0𝑓(πœ†)π‘˜0+2π‘˜0ξ€·π‘˜0ξ€Έβˆ’1𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1)βˆ’π‘˜0𝑓(πœ†)π‘˜0βˆ’2π‘˜0𝑓(πœ†βˆ’1)𝑓(πœ†)π‘˜0βˆ’1=π‘˜0𝑓(πœ†)π‘˜0+2π‘˜0ξ‚€π‘˜0βˆ’32𝑓(πœ†)π‘˜0βˆ’1𝑓(πœ†βˆ’1)>0.(4.5)
By Theorem 4.2, we obtain the order of the Hosoya index of 𝐢(𝑛,π‘˜0,πœ†).

Corollary 4.3. Let 3β‰€πœ†β‰€βŒŠ(π‘›βˆ’1)/π‘˜0βŒ‹+1, π‘˜0β‰₯1, 𝑛β‰₯5. Then 𝑧(𝐢(𝑛,π‘˜0,3))<𝑧(𝐢(𝑛,π‘˜0,4))<𝑧(𝐢(𝑛,π‘˜0,5))<β‹―, and 𝐢(𝑛,π‘˜0,3) has the smallest Hosoya index among the graphs in 𝐢(𝑛,π‘˜0,πœ†).

Theorem 4.4. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†0βˆ’1)βŒ‹, πœ†0β‰₯3, 𝑛β‰₯3. Then 𝑧(𝐢(𝑛,π‘˜,πœ†0))<𝑧(𝐢(𝑛,π‘˜+1,πœ†0)).

Proof. Let π‘˜β‰₯1, πœ†0β‰₯3, 𝑛β‰₯3, Ξ”6𝐢=𝑧𝑛,π‘˜+1,πœ†0ξ€·πΆξ€·ξ€Έξ€Έβˆ’π‘§π‘›,π‘˜,πœ†0=ξ€Ίξ€·πœ†ξ€Έξ€Έπ‘›βˆ’0ξ€Έξ€»π‘“ξ€·πœ†βˆ’1(π‘˜+1)0ξ€Έπ‘˜+1ξ€·πœ†+2(π‘˜+1)𝑓0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜βˆ’ξ€Ίξ€·πœ†π‘›βˆ’0ξ€Έπ‘˜ξ€»π‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜ξ€·πœ†βˆ’2π‘˜π‘“0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜βˆ’1=ξ‚ƒξ€·πœ†π‘›βˆ’0ξ€Έξ€·πœ†βˆ’1(π‘˜+1),𝑓0ξ€Έπ‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έπ‘˜ξ‚„βˆ’ξ€·πœ†0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜ξ€·πœ†+2π‘˜π‘“0ξ€Έξ‚ƒπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜ξ€·πœ†βˆ’π‘“0ξ€Έπ‘˜βˆ’1ξ‚„ξ€·πœ†+2𝑓0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜.(4.6)
Obviously, π‘›βˆ’(πœ†0βˆ’1)(π‘˜+1)β‰₯1 by 𝑧(𝐢(𝑛,π‘˜+1,πœ†0)) exists. We have ξ€Ίξ€·πœ†π‘›βˆ’0ξ€Έξ€»ξ‚ƒπ‘“ξ€·πœ†βˆ’1(π‘˜+1)0ξ€Έπ‘˜+1ξ€·πœ†βˆ’π‘“0ξ€Έπ‘˜ξ‚„ξ€·πœ†β‰₯0,2π‘˜π‘“0ξ€Έξ‚ƒπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜ξ€·πœ†βˆ’π‘“0ξ€Έπ‘˜βˆ’1ξ‚„>0.(4.7)
Thus Ξ”6ξ€·πœ†>2𝑓0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜βˆ’ξ€·πœ†0ξ€Έπ‘“ξ€·πœ†βˆ’10ξ€Έπ‘˜ξ€·πœ†=𝑓0ξ€Έπ‘˜ξ€Ίξ€·πœ†2𝑓0ξ€Έβˆ’ξ€·πœ†βˆ’10βˆ’1ξ€Έξ€»β‰₯0(byLemma2.1(ii)).(4.8)
By Theorem 4.4, we obtain the order of the Hosoya index of 𝐢(𝑛,π‘˜,πœ†0).

Corollary 4.5. Let 1β‰€π‘˜β‰€βŒŠ(π‘›βˆ’1)/(πœ†0βˆ’1)βŒ‹, πœ†0β‰₯3, 𝑛β‰₯3. Then 𝑧(𝐢(𝑛,1,πœ†0))<𝑧(𝐢(𝑛,2,πœ†0))<𝑧(𝐢(𝑛,2,πœ†0))<β‹―, and 𝐢(𝑛,1,πœ†0) has the smallest Hosoya index of among the graphs in 𝐢(𝑛,π‘˜,πœ†0).

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 Β· View at Google Scholar Β· View at 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, β€œTopological 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, β€œThe 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 Β· View at Zentralblatt MATH
  5. M. Fischermann, I. Gutman, A. Hoffmann, D. Rautenbach, D. Vidovic, and L. and Volkmann, β€œExtremal 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, β€œA 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 Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. I. Gutman, β€œExtremal hexagonal chains,” Journal of Mathematical Chemistry, vol. 12, no. 1–4, pp. 197–210, 1993. View at Publisher Β· View at Google Scholar
  8. I. Gutman and F. J. Zhang, β€œOn the ordering of graphs with respect to their matching numbers,” Discrete Applied Mathematics, vol. 15, no. 1, pp. 25–33, 1986. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. Y. Hou, β€œOn acyclic systems with minimal Hosoya index,” Discrete Applied Mathematics, vol. 119, no. 3, pp. 251–257, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. S. G. Wagner, β€œExtremal 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 Β· View at Zentralblatt MATH
  11. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, Germany, 1986. View at Publisher Β· View at Google Scholar
  12. J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, NY, USA, 1976.
  13. H. Deng, β€œThe 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 Β· View at Google Scholar
  14. H. Deng, β€œThe largest Hosoya index of (n,n+1)-graphs,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2499–2506, 2008. View at Publisher Β· View at Google Scholar