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

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 𝑁_𝐺(𝑣), and [𝑣]=𝑁_𝐺(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 𝑆_𝑛, 𝑃_𝑛, and 𝐶_𝑛 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.

Figure 1 

𝐶(𝑛,𝑘,𝜆) 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 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)≤2^𝑛−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,…,𝐺_𝑚 are the components of the graph 𝐺, then 𝑖(𝐺)=∏^𝑚𝑖=1𝑖(𝐺_𝑖) (see [10, Lemma 1]).(ii)If 𝑥∈𝑉(𝐺), then 𝑖(𝐺)=𝑖(𝐺−{𝑥})+𝑖(𝐺−[𝑥]) (see [10, Lemma 1]).(iii)𝑖(𝑆_𝑛)=2^𝑛−1+1; 𝑖(𝑃_𝑛)=𝑓(𝑛+2) for any 𝑛∈ℕ; 𝑖(𝐶_𝑛)=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,…,𝐺_𝑚 are the components of the graph 𝐺, then 𝑧(𝐺)=∏^𝑚𝑖=1𝑧(𝐺_𝑖) (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(𝑆_𝑛)=𝑛; 𝑧(𝑃_𝑛)=𝑓(𝑛+1) for any 𝑛∈ℕ; 𝑧(𝐶_𝑛)=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 𝑖(𝐶(𝑛,𝑘,𝜆))=𝑖(𝐺−{𝑣})+𝑖(𝐺−[𝑣])=𝑖𝑃₁^{[𝑛−(𝜆−1)𝑘−1]}𝑖(𝑃_𝜆−1)^𝑘+𝑖𝑃_𝜆−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==𝑖2𝐶𝑛,𝑘_{0(𝑛−𝜆𝑘0−1)},𝜆+1−𝑖𝐶𝑛,𝑘₀,𝜆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)^𝑘0−2^𝑘0𝑓(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)^𝑘0−2^𝑘0𝑓(𝜆+1)^𝑘0≤𝑓(𝜆+2)^𝑘0−2^𝑘0𝑓(𝜆+1)^𝑘0.(3.4) Thus Δ₁≤𝑓(𝜆+2)^𝑘0−2^𝑘0𝑓(𝜆+1)^𝑘0+𝑓(𝜆)^𝑘0−𝑓(𝜆−1)^𝑘0==[𝑓𝑓((𝜆𝜆++11))+_𝑘0𝑓(𝜆)]^𝑘0−2^𝑘0𝑓(𝜆+1)^𝑘0+𝑓(𝜆)^𝑘0−𝑓(𝜆−1)^𝑘0−2_𝑘0+𝑘₀𝑓(𝜆+1)^𝑘0−1𝑓(𝜆)+⋯+𝑓(𝜆)^𝑘0<2_𝑘0𝑓(𝜆+1)^𝑘0+𝑓(𝜆)^𝑘0−𝑓(𝜆−1)^𝑘0=−𝑓(𝜆−+11)𝑓_𝑘0(𝜆+1)^𝑘0+𝑓(𝜆)^𝑘0−2^𝑘0𝑓(𝜆+1)^𝑘0+𝑓(𝜆)^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+2^𝑘0−1𝑓(𝜆−1)^𝑘0+2𝑓(𝜆)^𝑘0−𝑓(𝜆−1)^𝑘0==−−2(^𝑘02𝑓𝑓(𝜆(𝜆−−11)))^𝑘0+_𝑘0𝑓(𝜆)^𝑘0(by𝑓(𝜆)<2𝑓(𝜆−1))−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==𝑖2(𝐶(𝑛,𝑘+_{[𝑛−2(𝑘+1)−1]}1,3))−𝑖(𝐶(𝑛,𝑘,3))𝑓(4)^𝑘+1+𝑓(2)^𝑘+1−2^{[𝑛−2𝑘−1]}𝑓(4)^𝑘−𝑓(2)^𝑘=2^𝑛−134^𝑘+1−2^𝑛−134^𝑘=−2^𝑛−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=2^{𝑛−3𝑘−4}5^𝑘(−3)+2^𝑘≤−3⋅5^𝑘+2^𝑘<0.(3.7) If 𝜆₀≥5, then Δ₄==2𝑖𝐶𝑛,𝑘+1,𝜆_{0[𝑛−(𝜆0−1)(𝑘+1)−1]}−𝑖𝐶𝑛,𝑘,𝜆₀𝑓𝜆₀+1^𝑘+1+𝑓𝜆₀−1^𝑘+1−2^{[𝑛−(𝜆0−1)𝑘−1]}𝑓𝜆₀+1^𝑘−𝑓𝜆₀−1^𝑘=2+𝑓𝜆^{[𝑛−(𝜆0−1)(𝑘+1)−1]}₀𝑓𝜆₀+1^𝑘𝑓𝜆₀+1−2^𝜆0−1−1^𝑘+1−𝑓𝜆₀−1^𝑘.(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−2^𝜆0−1≤−𝑓𝜆₀+1≤0.(3.9) Thus Δ₄≤𝑓𝜆₀+1^𝑘𝑓𝜆₀+1−2^𝜆0−1+𝑓𝜆₀−1^𝑘+1−𝑓𝜆₀−1^𝑘≤𝑓𝜆₀+1^𝑘−𝑓𝜆₀+1+𝑓𝜆₀−1^𝑘+1−𝑓𝜆₀−1^𝑘=−𝑓𝜆₀+1^𝑘+1−𝑓𝜆₀−1^𝑘+1−𝑓𝜆⁰−1^𝑘<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)𝑘]𝑓(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𝑧((𝐺𝑃−_𝜆−1{𝑣})+_{𝑥∈𝑁𝐺(𝑣)}𝑧(𝐺−{𝑥,𝑣}))^𝑘+[𝑛−(𝜆−1)𝑘−1]𝑧(𝑃_𝜆−1)^𝑘+2𝑘𝑧(𝑃_𝜆−2),𝑧(𝑃_𝜆−1)^𝑘−1=𝑓(𝜆)^𝑘+[𝑛−(𝜆−1)𝑘−1]𝑓(𝜆)^𝑘+2𝑘𝑓(𝜆−1)𝑓(𝜆)^𝑘−1=[𝑛−(𝜆−1)𝑘]𝑓(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==𝑧𝑛𝐶−𝜆𝑛𝑘,₀𝑘₀,𝜆+1−𝑧𝐶𝑛,𝑘₀,𝜆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(𝑓𝜆(𝜆++1)1)^𝑘0−_𝑘0𝑓(𝜆)^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)^𝑘02𝑘₀=𝑘₀𝑓(𝜆)^𝑘0−1𝑓(𝜆−1),𝑓(𝜆)𝑓(𝜆+1)^𝑘0−1==22𝑘𝑘₀₀𝑓(𝜆)[𝑓(𝜆)+𝑓(𝜆−1)]^𝑘0−1≥2𝑘₀𝑓(𝜆)𝑓(𝜆)^𝑘0−1+𝑘₀−1𝑓(𝜆)^𝑘0−2𝑓(𝜆−1)𝑓(𝜆)^𝑘0+2𝑘₀𝑘₀−1𝑓(𝜆)^𝑘0−1𝑓(𝜆−1).(4.4)
Thus Δ₅≥−𝑘₀𝑘𝑓₀(𝜆)^𝑘0−1𝑓(𝜆−1)+2𝑘₀𝑓(𝜆)^𝑘0+2𝑘₀𝑘₀−1𝑓(𝜆)^𝑘0−1𝑓(𝜆−1)𝑓(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==𝑧𝑛−𝐶𝜆𝑛,₀𝑘+1,𝜆₀−𝑧𝐶𝑛,𝑘,𝜆₀−1(𝑘+1)𝑓𝜆₀^𝑘+1+2(𝑘+1)𝑓𝜆₀−1𝑓𝜆₀^𝑘−𝑛−𝜆₀−1𝑘𝑓𝜆₀^𝑘−2𝑘𝑓𝜆₀−1𝑓𝜆₀^𝑘−1=𝑛−𝜆₀−1(𝑘+1),𝑓𝜆₀^𝑘+1−𝑓𝜆₀^𝑘−𝜆⁰−1𝑓𝜆₀^𝑘+2𝑘𝑓𝜆₀−1𝑓𝜆₀^𝑘−𝑓𝜆₀^𝑘−1+2𝑓𝜆⁰−1𝑓𝜆₀^𝑘.(4.6)
Obviously, 𝑛−(𝜆₀−1)(𝑘+1)≥1 by 𝑧(𝐶(𝑛,𝑘+1,𝜆₀)) exists. We have 𝑛−2𝑘𝜆𝑓⁰−𝜆1₀(𝑘+1)𝑓𝜆₀^𝑘+1−𝑓𝜆₀^𝑘≥0,−1𝑓𝜆₀^𝑘−𝑓𝜆₀^𝑘−1>0.(4.7)
Thus Δ₆>2𝑓𝜆₀−1𝑓𝜆₀^𝑘−𝜆₀−1𝑓𝜆₀^𝑘=𝑓𝜆₀^𝑘2𝑓𝜆₀−1−𝜆₀−1≥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.