Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 520156, 8 pages
doi:10.1155/2012/520156
Research Article
The Merrifield-Simmons Index and Hosoya Index
of Cn, k, λ Graphs
Shaojun Dai1 and Ruihai Zhang2
1
Department of Mathematics, Tianjin Polytechnic University, No. 399 Binshuixi Road, Xiqing District,
Tianjin 300387, China
2
Department of Mathematics, Tianjin University of Science and Technology, Tianjin 300457, China
Correspondence should be addressed to Shaojun Dai, daishaojun1978@sina.com
Received 29 May 2012; Revised 18 June 2012; Accepted 27 June 2012
Academic Editor: Ferenc Hartung
Copyright q 2012 S. Dai and R. 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.
The Merrifield-Simmons index iG 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 zG of a graph G is defined as the total number of independent edge subsets,
that is, the total number of its matchings. By Cn, 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 iG
and the Hosoya index zG for a graph G in Cn, k, λ.
1. Introduction
Let G V G, EG denote a graph whose set of vertices and set of edges are V G and
EG, 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
Cn, 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 − 1k 1 ≥ 0 and the vertex v denotes the central vertex of the graphs, as
shown in Figure 1. The Merrifield-Simmons index iG 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 zG of a graph G is defined as the total
2
Journal of Applied Mathematics
λ−1
k
V
n1
Figure 1: Cn, k, λ graphs.
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 iG and the Hosoya index
zG for a graph G in Cn, 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 ⊆ EG 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 fn we denote the nth Fibonacci number, where n ∈ N,
fn fn 1 fn 2 with initial conditions f0 0 and f1 1.
The following lemma is obvious.
Lemma 2.1. Let n ∈ N.
i If n ≥ 6, then fn ≤ 2n−3 .
ii If n ≥ 2, then fn ≥ n/2.
We will make use of the following two well-known lemmas on the MerrifieldSimmons index and Hosoya index.
Lemma 2.2. Let G V G, EG be a graph.
i If G1 , G2 , . . . , Gm are the components of the graph G, then iG Lemma 1]).
m
i1 iGi (see [10,
ii If x ∈ V G, then iG iG − {x} iG − x (see [10, Lemma 1]).
iii iSn 2n−1 1; iPn fn 2 for any n ∈ N; iCn fn − 1 fn 1 for any
n ≥ 3 (see [13]).
Journal of Applied Mathematics
3
Lemma 2.3. Let G V G, EG be a graph.
i If G1 , G2 , . . . , Gm are the components of the graph G, then zG Lemma 1]).
m
i1 zGi (see [10,
ii If e xy ∈ EG, then zG zG − {e} zG − {x, y} (see [14]).
iii If x ∈ V G, then zG zG − {x} y∈NG x
zG − {x, y} (see [10, Lemma 1]).
iv zSn n; zPn fn 1 for any n ∈ N; zCn fn − 1 fn 1 for any n ≥ 3
(see [14]).
3. The Merrifield-Simmons Index of Cn, k, λ
In this section, we will give the Merrifield-Simmons index of Cn, k, λ and their order.
Theorem 3.1. Let 1 ≤ k ≤ n − 1/λ − 1, λ ≥ 3. Then
iCn, k, λ 2n−λ−1k−1 fλ 1k fλ − 1k .
3.1
Proof. By Lemma 2.2 and an elementary calculating, we have
iCn, k, λ iG − {v} iG − v iP1 n−λ−1k−1 iPλ−1 k iPλ−3 k
2n−λ−1k−1 fλ 1k fλ − 1k .
3.2
Theorem 3.2. Let 3 ≤ λ ≤ n − 1/k0 1, k0 ≥ 1, n ≥ 5. Then iCn, k0 , λ 1 < iCn, k0 , λ.
Proof. Let 3 ≤ λ ≤ n − 1/k0 1. We have
Δ1 iCn, k0 , λ 1 − iCn, k0 , λ
2n−λk0 −1 fλ 2k0 fλk0 − 2n−k0 λ−1−1 fλ 1k0 − fλ − 1k0
2n−λk0 −1 fλ 2k0 − 2k0 fλ 1k0 fλk0 − fλ − 1k0 .
3.3
Obviously, 2n−λk0 −1 ≥ 1 by zCn, k0 , λ 1 be exist. Again by fλ 2 < 2fλ 1, we
have
2n−λk0 −1 fλ 2k0 − 2k0 fλ 1k0 ≤ fλ 2k0 − 2k0 fλ 1k0 .
3.4
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Journal of Applied Mathematics
Thus
Δ1 ≤ fλ 2k0 − 2k0 fλ 1k0 fλk0 − fλ − 1k0
k
fλ 1 fλ 0 − 2k0 fλ 1k0 fλk0 − fλ − 1k0
fλ 1k0 k0 fλ 1k0 −1 fλ · · · fλk0
− 2k0 fλ 1k0 fλk0 − fλ − 1k0
<
2k0 − 1 fλ 1k0 fλk0 − 2k0 fλ 1k0 fλk0 − fλ − 1k0
3.5
−fλ 1k0 2fλk0 − fλ − 1k0
k
− fλ fλ − 1 0 2fλk0 − fλ − 1k0
< − fλk0 2k0 − 1 fλ − 1k0 2fλk0 − fλ − 1k0
−2k0 fλ − 1k0 fλk0
byfλ < 2fλ − 1
k
− 2fλ − 1 0 − fλk0 < 0.
By Theorem 3.2, we obtain the order of the Merrifield-Simmons index of Cn, k0 , λ.
Corollary 3.3. Let 3 ≤ λ ≤ n − 1/k0 1, k0 ≥ 1, n ≥ 5. Then iCn, k0 , 3 > iCn, k0 , 4 >
iCn, k0 , 5 > · · · , and Cn, k0 , 3 has the largest Merrifield-Simmons index among the graphs in
Cn, k0 , λ.
Theorem 3.4. Let 1 ≤ k ≤ n−1/λ0 −1, λ0 ≥ 3, n ≥ 3. Then iCn, k 1, λ0 < iCn, k, λ0 .
Proof. Let k ≥ 1, n ≥ 3. If λ0 3, then
Δ2 iCn, k 1, 3 − iCn, k, 3
2n−2k1−1 f4k1 f2k1 − 2n−2k−1 f4k − f2k
k1
k
k
1
3
3
3
2n−1
− 2n−1
−2n−1
· < 0.
4
4
4
4
3.6
If λ0 4, then
Δ3 iCn, k 1, 4 − iCn, k, 4 2n−3k1−1 f5k1 f3k1
− 2n−3k−1 f5k − f3k 2n−3k−4 5k −3 2k ≤ −3 · 5k 2k < 0.
3.7
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5
If λ0 ≥ 5, then
Δ4 iCn, k 1, λ0 − iCn, k, λ0 2n−λ0 −1k1−1 fλ0 1k1 fλ0 − 1k1
− 2n−λ0 −1k−1 fλ0 1k − fλ0 − 1k
2n−λ0 −1k1−1 fλ0 1k fλ0 1 − 2λ0 −1
3.8
fλ0 − 1k1 − fλ0 − 1k .
Obviously, 2n−λ0 −1k1−1 ≥ 1 by zCn, k 1, λ0 exists. Again by Lemma 2.1i, we have
fλ0 1 − 2λ0 −1 ≤ −fλ0 1 ≤ 0.
3.9
Thus
Δ4 ≤ fλ0 1k fλ0 1 − 2λ0 −1 fλ0 − 1k1 − fλ0 − 1k
≤ fλ0 1k −fλ0 1 fλ0 − 1k1 − fλ0 − 1k
− fλ0 1k1 − fλ0 − 1k1 − fλ0 − 1k < 0.
3.10
By Theorem 3.4, we obtain the order of the Merrifield-Simmons index of Cn, k, λ0 .
Corollary 3.5. Let 1 ≤ k ≤ n − 1/λ0 − 1, n ≥ 3. Then iCn, 1, λ0 > iCn, 2, λ0 >
iCn, 3, λ0 > · · · , and Cn, 1, λ0 has the largest Merrifield-Simmons index of among the graphs in
Cn, k, λ0 .
4. The Hosoya Index of Cn, k, λ
In this section, we will give the Hosoya index of Cn, k, λ and their order.
Theorem 4.1. Let 1 ≤ k ≤ n − 1/λ − 1 and λ ≥ 3. Then
zCn, k, λ n − λ − 1kfλk 2kfλ − 1fλk−1 .
4.1
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Journal of Applied Mathematics
Proof. For all 1 ≤ k ≤ n − 1/λ − 1 and λ ≥ 3, according to Lemma 2.3, we have the
following:
zCn, k, λ zG − {v} zG − {x, v}
x∈NG v
zPλ−1 k n − λ − 1k − 1zPλ−1 k 2kzPλ−2 , zPλ−1 k−1
4.2
fλk n − λ − 1k − 1fλk 2kfλ − 1fλk−1
n − λ − 1kfλk 2kfλ − 1fλk−1 .
Theorem 4.2. Let 3 ≤ λ ≤ n − 1/k0 1, k0 ≥ 1 and n ≥ 5. Then zCn, k0 , λ < zCn, k0 , λ 1.
Proof. Let 3 ≤ λ ≤ n − 1/k0 1, k0 ≥ 1 and n ≥ 5. We have
Δ5 zCn, k0 , λ 1 − zCn, k0 , λ
n − λk0 fλ 1k0 2k0 fλfλ 1k0 −1
− n − λk0 k0 fλk0 − 2k0 fλ − 1fλk0 −1
n − λk0 fλ 1k0 − fλk0 2k0 fλfλ 1k0 −1
4.3
− k0 fλk0 − 2k0 fλ − 1fλk0 −1 .
Obviously, n − λk0 ≥ 1 by zCn, k0 , λ 1 exists. We have
n − λk0 fλ 1k0 − fλk0
≥ fλ 1k0 − fλk0
k −1
fλ fλ − 1 0 − fλk0
≥ fλk0 k0 fλk0 −1 fλ − 1 − fλk0
k0 fλk0 −1 fλ − 1,
2k0 fλfλ 1k0 −1
k −1
2k0 fλ fλ fλ − 1 0 ≥ 2k0 fλ fλk0 −1 k0 − 1fλk0 −2 fλ − 1
2k0 fλk0 2k0 k0 − 1fλk0 −1 fλ − 1.
4.4
Journal of Applied Mathematics
7
Thus
Δ5 ≥ k0 fλk0 −1 fλ − 1 2k0 fλk0 2k0 k0 − 1fλk0 −1 fλ − 1
− k0 fλk0 − 2k0 fλ − 1fλk0 −1
3
k0
fλk0 −1 fλ − 1 > 0.
k0 fλ 2k0 k0 −
2
4.5
By Theorem 4.2, we obtain the order of the Hosoya index of Cn, k0 , λ.
Corollary 4.3. Let 3 ≤ λ ≤ n − 1/k0 1, k0 ≥ 1, n ≥ 5. Then zCn, k0 , 3 < zCn, k0 , 4 <
zCn, k0 , 5 < · · · , and Cn, k0 , 3 has the smallest Hosoya index among the graphs in Cn, k0 , λ.
Theorem 4.4. Let 1 ≤ k ≤ n−1/λ0 −1, λ0 ≥ 3, n ≥ 3. Then zCn, k, λ0 < zCn, k1, λ0 .
Proof. Let k ≥ 1, λ0 ≥ 3, n ≥ 3,
Δ6 zCn, k 1, λ0 − zCn, k, λ0 n − λ0 − 1k 1fλ0 k1 2k 1fλ0 − 1fλ0 k
− n − λ0 − 1kfλ0 k − 2kfλ0 − 1fλ0 k−1
n − λ0 − 1k 1, fλ0 k1 − fλ0 k − λ0 − 1fλ0 k
4.6
2kfλ0 − 1 fλ0 k − fλ0 k−1 2fλ0 − 1fλ0 k .
Obviously, n − λ0 − 1k 1 ≥ 1 by zCn, k 1, λ0 exists. We have
n − λ0 − 1k 1 fλ0 k1 − fλ0 k ≥ 0,
2kfλ0 − 1 fλ0 k − fλ0 k−1 > 0.
4.7
Thus
Δ6 > 2fλ0 − 1fλ0 k − λ0 − 1fλ0 k
fλ0 k 2fλ0 − 1 − λ0 − 1 ≥ 0 by Lemma 2.1ii .
4.8
By Theorem 4.4, we obtain the order of the Hosoya index of Cn, k, λ0 .
Corollary 4.5. Let 1 ≤ k ≤ n − 1/λ0 − 1, λ0 ≥ 3, n ≥ 3. Then zCn, 1, λ0 <
zCn, 2, λ0 < zCn, 2, λ0 < · · · , and Cn, 1, λ0 has the smallest Hosoya index of among the
graphs in Cn, k, λ0 .
8
Journal of Applied Mathematics
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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