International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 282940 | 10 pages | https://doi.org/10.5402/2011/282940

Some Results on the Signless Laplacian Spectra of Unicyclic Graphs

Academic Editor: M. Asaad
Received06 Jun 2011
Accepted22 Jun 2011
Published25 Aug 2011

Abstract

We determine the second to fourth largest (resp. the second smallest) signless Laplacian spectral radii and the second to fourth largest signless Laplacian spreads together with the corresponding graphs in the class of unicyclic graphs with n vertices. Moreover, we prove that one class of unicyclic graphs are determined by their signless Laplacian spectra.

1. Introduction

Throughout the paper, 𝐺=(𝑉,𝐸) is an undirected simple graph with 𝑛 vertices and 𝑚 edges. If 𝐺 is connected with 𝑚=𝑛+𝑐−1, then 𝐺 is called a 𝑐-cyclic graph. Especially, if 𝑐=0 or 1, then 𝐺 is called a tree or a unicyclic graph, respectively. Let 𝕌𝑛 be the class of unicyclic graphs with 𝑛 vertices. The neighbor set of a vertex 𝑣 is denoted by 𝑁(𝑣). We write 𝑑(𝑣) for the degree of vertex 𝑣. In particular, let Δ(𝐺) and 𝛿(𝐺) be the maximum degree and minimum degree of 𝐺, respectively. Let 𝐴(𝐺) be the adjacency matrix and 𝐷(𝐺) be the diagonal matrix whose (𝑖,𝑖)-entry is 𝑑(𝑣𝑖), of 𝐺, respectively. The signless Laplacian matrix of 𝐺 is 𝑄(𝐺)=𝐷(𝐺)+𝐴(𝐺). Clearly, 𝑄(𝐺) is positive semidefinite [1] and its eigenvalues can be arranged as𝜇1(𝐺)≥𝜇2(𝐺)≥⋯≥𝜇𝑛(𝐺)≥0.(1.1) Let 𝜇(𝐺) be the signless Laplacian spectral radius of 𝐺, namely, 𝜇(𝐺)=𝜇1(𝐺). The signless Laplacian spread of 𝐺 is defined as 𝑆𝑄(𝐺)=𝜇1(𝐺)−𝜇𝑛(𝐺) [1, 2]. Let Φ(𝐺,𝑥) be the signless Laplacian characteristic polynomial of 𝐺, that is, Φ(𝐺,𝑥)=det(𝑥𝐼−𝑄(𝐺)).

Recently, the research on the spectrum of 𝑄(𝐺) receive much attention. Some properties of signless Laplacian spectra of graphs and some possibilities for developing the spectral theory of graphs based on 𝑄(𝐺) are discussed in [3–5]. The largest signless Laplacian spectral radius and the largest signless Laplacian spread among the class of unicyclic graphs with 𝑛 vertices were determined in [6] and [1], respectively. The smallest signless Laplacian spectral radius among the class of unicyclic graphs with 𝑛 vertices was determined in [7]. In this paper, we will determine the second to fourth largest and the second smallest signless Laplacian spectral radii together with the corresponding graphs in the class of unicyclic graphs with 𝑛 vertices. Moreover, we also indentify the second to fourth largest signless Laplacian spreads together with the corresponding graphs in the class of unicyclic graphs with 𝑛 vertices. In the end of this paper, we will prove that a class of unicyclic graphs are determined by their signless Laplacian spectra.

2. The Signless Laplacian Spectral Radii of Unicyclic Graphs

As usually, let 𝐾1,𝑛−1, 𝑃𝑛, and 𝐶𝑛 be the star, path, and cycle with 𝑛 vertices, respectively. In the following, let 𝑆3𝑛(𝑛≥4) be the unicyclic graph obtained by adding one edge to two pendant vertices of 𝐾1,𝑛−1, and let 𝐹𝑛, 𝐻𝑛, 𝑆4𝑛 be the unicyclic graphs with 𝑛 vertices as shown in Figure 1.

In [6], the largest signless Laplacian spectral radius in the class of unicyclic graphs was determined, and it was proved as follows.

Theorem 2.1 (see [6]). If 𝑈∈𝕌𝑛, and 𝑛≥4, then 𝜇(𝑈)≤𝜇(𝑆3𝑛), where the equality holds if and only if 𝑈≅𝑆3𝑛.

Theorem 2.2. Suppose 𝑈∈𝕌𝑛, and 𝑛≥8. (1) If 𝑈≇𝑆3𝑛, then 𝜇(𝑈)≤𝜇(𝐹𝑛), where the equality holds if and only if 𝑈≅𝐹𝑛, and 𝜇(𝐹𝑛) equals the maximum root of the equation 𝑥5−(𝑛+5)𝑥4+(6𝑛+3)𝑥3−(9𝑛−1)𝑥2+(3𝑛+8)𝑥−4=0. (2) If 𝑈≇𝑆3𝑛 and 𝑈≇𝐹𝑛, then 𝜇(𝑈)≤𝜇(𝐻𝑛), where the equality holds if and only if 𝑈≅𝐻𝑛, and 𝜇(𝐻𝑛) equals the maximum root of the equation 𝑥5−(𝑛+5)𝑥4+(6𝑛+4)𝑥3−(10𝑛−2)𝑥2+(3𝑛+12)𝑥−4=0. (3) If 𝑈≇𝑆3𝑛, 𝑈≇𝐹𝑛 and 𝑈≇𝐻𝑛, then 𝜇(𝑈)≤𝜇(𝑆4𝑛), where the equality holds if and only if 𝑈≅𝑆4𝑛, and 𝜇(𝑆4𝑛) equals the maximum root of the equation 𝑥3−(𝑛+3)𝑥2+(4𝑛−2)𝑥−2𝑛=0.

In order to prove Theorem 2.2, the following lemmas are needed.

Lemma 2.3 (see [8]). 𝜇(𝐺)≤max{𝑑(𝑣)+𝑚(𝑣),𝑣∈𝑉(𝐺)}, where ∑𝑚(𝑣)=𝑢∼𝑣𝑑(𝑢)/𝑑(𝑣).

Proposition 2.4. Suppose 𝑐≥1 and 𝐺 is a 𝑐-cyclic graph on 𝑛 vertices with Δ≤𝑛−3. If 𝑛≥2𝑐+5, then 𝜇(𝐺)≤𝑛−1.

Proof. We only need to prove that max{𝑑(𝑣)+𝑚(𝑣)∶𝑣∈𝑉}≤𝑛−1 by Lemma 2.3. Suppose 𝑑(𝑢)+𝑚(𝑢)=max{𝑑(𝑣)+𝑚(𝑣)∶𝑣∈𝑉}. We consider the next three cases.
Case 1 (𝑑(𝑢)=1). Suppose 𝑣∈𝑁(𝑢). Then, 𝑑(𝑢)+𝑚(𝑢)=1+𝑑(𝑣)≤1+Δ≤𝑛−2<𝑛−1.
Case 2 (𝑑(𝑢)=2). Suppose that 𝑣,𝑤∈𝑁(𝑢). Then, 𝑑(𝑢)+𝑚(𝑢)=2+𝑑(𝑣)+𝑑(𝑤)2≤2+2Δ2=Δ+2≤𝑛−1.(2.1)
Case 3 (3≤𝑑(𝑢)≤𝑛−3). Note that 𝐺 has 𝑛+𝑐−1 edges and 3≤𝑑(𝑢)≤𝑛−3. Then, 𝑑(𝑢)+𝑚(𝑢)≤𝑑(𝑢)+2(𝑛+𝑐−1)−𝑑(𝑢)−2𝑑(𝑢)=𝑑(𝑢)−1+2𝑛+2𝑐−4𝑑(𝑢).(2.2) Next we will prove that 𝑑(𝑢)−1+((2𝑛+2𝑐−4)/𝑑(𝑢))≤𝑛−1, equivalently, 𝑑(𝑢)(𝑛−𝑑(𝑢))≥2𝑛+2𝑐−4. Let 𝑓(𝑥)=(𝑛−𝑥)𝑥, where 3≤𝑥≤𝑛−3. Since 𝑓′(𝑥)=𝑛−2𝑥 and 3≤𝑥≤𝑛−3, we have 𝑓(𝑥)≥min{𝑓(3),𝑓(𝑛−3)}=3(𝑛−3)≥2𝑛+2𝑐−4.
By combining the above arguments, the result follows.

Corollary 2.5. Suppose 𝑈∈𝕌𝑛. If 𝑛≥7 and Δ≤𝑛−3, then 𝜇(𝑈)≤𝑛−1.

Lemma 2.6 (see [6]). If 𝐺 is a connected graph of order 𝑛≥4, then 𝜇(𝐺)≥Δ+1, where the equality holds if and only if 𝐺≅𝐾1,𝑛−1.

Suppose 𝐵 is a square matrix, let ğ‘Žğ‘–ğ‘–(𝐵) be the entry appearing in the 𝑖th row and the 𝑖th column of 𝐵. The next result gives a new method to calculate the signless Lapalacian characteristic polynomial of an 𝑛-vertex graph via the aid of computer.

Lemma 2.7 (see [9]). Let 𝐺 be a graph on 𝑛−𝑘 (1≤𝑘≤𝑛−2) vertices with 𝑉(𝐺)={𝑣𝑛,𝑣𝑛−1,…,𝑣𝑘+1}. If 𝐺′ is obtained from 𝐺 by attaching 𝑘 new pendant vertices, say 𝑣1,…,𝑣𝑘, to 𝑣𝑘+1, then Î¦î€·ğ‘„î€·ğºî…žî€¸î€¸,𝑥=(𝑥−1)𝑘⋅det𝑥𝐼𝑛−𝑘−𝑄(𝐺)−𝐵𝑛−𝑘,(2.3) where ğ‘Ž11(𝑄(𝐺)) is corresponding to the vertex 𝑣𝑘+1, and 𝐵𝑛−𝑘=diag{𝑘+(𝑘/(𝑥−1)),0,…,0}.

Example 2.8. Let 𝐹𝑛 be the unicyclic graph as shown in Figure 1. By Lemma 2.7, we have Φ𝐹𝑛,𝑥=(𝑥−1)𝑛−4⎛⎜⎜⎜⎜⎜⎝det(𝐵),where𝐵=𝑥−(𝑛−2)−𝑛−4âŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽ ğ‘¥âˆ’1−1−10−1𝑥−2−10−1−1𝑥−3−100−1𝑥−1.(2.4) By using “Matlab”, it easily follows that Φ𝐹𝑛,𝑥=(𝑥−1)𝑛−5𝑥5−(𝑛+5)𝑥4+(6𝑛+3)𝑥3−(9𝑛−1)𝑥2+(3𝑛+8)𝑥−4.(2.5) With the similar method, by Lemma 2.7 we have Φ𝐻𝑛,𝑥=(𝑥−1)𝑛−5𝑥5−(𝑛+5)𝑥4+(6𝑛+4)𝑥3−(10𝑛−2)𝑥2,Φ𝑆+(3𝑛+12)𝑥−4(2.6)4𝑛,𝑥=𝑥(𝑥−1)𝑛−5𝑥(𝑥−2)3−(𝑛+3)𝑥2+.(4𝑛−2)𝑥−2𝑛(2.7)

Proof of Theorem 2.2. Note that 𝑆3𝑛 is the unique unicyclic graph with Δ=𝑛−1, and 𝐹𝑛, 𝐻𝑛, 𝑆4𝑛 are all the unicyclic graphs with Δ=𝑛−2. Now suppose 𝑈∈𝕌𝑛⧵{𝑆3𝑛,𝐹𝑛,𝐻𝑛,𝑆4𝑛}. By Lemma 2.6 and Corollary 2.5, we have 𝜇(𝑆4𝑛)>𝑛−1≥𝜇(𝑈) because Δ(𝑈)≤𝑛−3.
To finish the proof of Theorem 2.2, we only need to show that 𝜇(𝑆4𝑛)<𝜇(𝐻𝑛)<𝜇(𝐹𝑛) by Theorem 2.1. By Lemma 2.6, it follows that 𝜇(𝐹𝑛)>𝑛−1,𝜇(𝐻𝑛)>𝑛−1, and 𝜇(𝑆4𝑛)>𝑛−1. When 𝑥≥𝑛−1 and 𝑛≥8, by (2.5), (2.6) and (2.7), it follows that Φ𝐻𝑛𝐹,𝑥−Φ𝑛,𝑥=(𝑥−1)𝑛−5𝑥𝑥2Φ𝑆−(𝑛−1)𝑥+4>0,4𝑛𝐻,𝑥−Φ𝑛=,𝑥(𝑥−1)𝑛−52𝑥2+≥(𝑛−12)𝑥+4(𝑥−1)𝑛−5(𝑥(3𝑛−14)+4)>0.(2.8) Therefore, we have 𝜇(𝐹𝑛)>𝜇(𝐻𝑛)>𝜇(𝑆4𝑛). Thus, Theorem 2.2 follows.

In [7], the smallest signless Laplacian spectral radius among all unicyclic graphs with 𝑛 vertices was determined, and that is as follows.

Theorem 2.9 (see [7]). If 𝑈∈𝕌𝑛, then 𝜇(𝑈)≥4, where the equality holds if and only if 𝑈≅𝐶𝑛.

The lollipop graph, denoted by 𝑊𝑛,𝑝, is obtained by appending a cycle 𝐶𝑝 to a pendant vertex of a path 𝑃𝑛−𝑝. The next result extends the order of Theorem 2.9.

Theorem 2.10. For any 𝑛, if 𝑈∈𝕌𝑛⧵{𝐶𝑛,𝑊𝑛,𝑛−1}, then 𝜇(𝑈)>𝜇(𝑊𝑛,𝑛−1)>𝜇(𝐶𝑛).

To prove Theorem 2.10, we will introduce more useful lemmas and notations.

Let 𝐺 be a connected graph, and 𝑢𝑣∈𝐸(𝐺). The graph 𝐺𝑢,𝑣 is obtained from 𝐺 by subdividing the edge 𝑢𝑣, that is, adding a new vertex 𝑤 and edges 𝑤𝑢,𝑤𝑣 in 𝐺−𝑢𝑣. An internal path, say 𝑣1𝑣2⋯𝑣𝑠+1(𝑠≥1), is a path joining 𝑣1 and 𝑣𝑠+1 (which need not be distinct) such that 𝑣1 and 𝑣𝑠+1 have degree greater than 2, while all other vertices 𝑣2,…,𝑣𝑠 are of degree 2.

Lemma 2.11 (see [3, 10]). Let 𝑢𝑣 be an edge of a connected graph 𝐺. If 𝑢𝑣 belongs to an internal path of 𝐺, then 𝜇(𝐺)>𝜇(𝐺𝑢,𝑣).

By 𝐺⊂𝐺′, we mean that 𝐺 is a subgraph of 𝐺′ and 𝐺≇𝐺′.

Lemma 2.12 (see [10]). If 𝐺⊂𝐺′ and 𝐺′ is a connected graph, then 𝜇(𝐺)<𝜇(ğºî…ž).

If 𝑢∈𝑉(𝐺) and 𝑑(𝑢)≥3, then we called 𝑢 a branching point of 𝐺. Let 𝑈 be a connected unicyclic graph and 𝑇𝑣 be a tree such that 𝑇𝑣 is attached to a vertex 𝑣 of the unique cycle of 𝑈. The vertex 𝑣 is called the root of 𝑇𝑣, and 𝑇𝑣 is called a root tree of 𝑈. Throughout this paper, we assume that 𝑇𝑣 does not include the root 𝑣. Clearly, 𝑈 is obtained by attaching root trees to some vertices of the unique cycle of 𝑈.

Proof of Theorem 2.10. In the proof of this result, we assume that the unique cycle of 𝑈 is 𝐶𝑝. Now choose 𝑈∈𝕌𝑛⧵{𝐶𝑛} such that 𝜇(𝑈) is as small as possible. Since 𝑈≇𝐶𝑛, 𝑈 has at least one branching point. We consider the next two cases.
Case 1. There are at least two branching points in 𝐶𝑝.
Now suppose 𝑢 and 𝑣 are two branching points in 𝐶𝑝 such that there does not exsit other barnching point between the shortest path in 𝐶𝑝 connected 𝑢 and 𝑣. Let 𝑢=𝑣1𝑣2⋯𝑣𝑠=𝑣(𝑠≥2) be the shortest path in 𝐶𝑝 connected 𝑢 and 𝑣. Since 𝑢 is a branching point of 𝑈, there is at least one pendant vertex, say 𝑤, in 𝑇𝑢. Suppose 𝑧 is the unique neighbor vertex of 𝑤 in 𝑈 (may be 𝑧=𝑢). Let 𝑈1=𝑈−𝑤𝑧 and 𝑈2=𝑈1−𝑤. Then, 𝑈2⊂𝑈, and hence 𝜇(𝑈2)<𝜇(𝑈) by Lemma 2.12. Let 𝑈3=𝑈1−𝑣1𝑣2+𝑣1𝑤+𝑣2𝑤. By the hypothesis, 𝑣1𝑣2⋯𝑣𝑠 is an internal path of 𝑈2. By Lemma 2.11, we can conclude that 𝜇(𝑈3)<𝜇(𝑈2)<𝜇(𝑈). But 𝑈3 is also a unicyclic graph with 𝑛 vertices and 𝑈3≇𝐶𝑛 because 𝑣 is a branching point of 𝑈3, it is a contradiction to the choice of 𝑈. Thus, Case 1 is impossible.
Case 2. There is unique branching point in 𝐶𝑝.
Subcase 1. There is at least a branching point outside 𝐶𝑝.
It can be proved analogously with Case 1.

Subcase 2. There does not exist any branching point outside 𝐶𝑝.Suppose 𝑢 is the unique brancing poing in 𝐶𝑝. By the hypothesis, 𝑢 is also the unique branching point of 𝑈. Then, 𝑈 is obtained by attaching 𝑑(𝑢)−2 paths to the vertex 𝑢 of 𝐶𝑝.
If 𝑑(𝑢)≥4, then there are at least two paths being attaching to 𝑢. It can be proved analogously with Case 1.
If 𝑑(𝑢)=3, then 𝑈 is a lollipop graph, that is, 𝑈≅𝑊𝑛,𝑝. Let 𝑉(𝑊𝑛,𝑝)={𝑣1,𝑣2,…,𝑣𝑛} and 𝐸(𝑊𝑛,𝑝)={𝑣𝑝𝑣1,𝑣𝑖𝑣𝑖+1,1≤𝑖≤𝑛−1}. If 𝑛−𝑝≥2, since 𝑊𝑛,𝑝−𝑣𝑛−1𝑣𝑛⊂𝑊𝑛,𝑝, 𝜇(𝑊𝑛,𝑝)>𝜇(𝑊𝑛,𝑝−𝑣𝑛−1𝑣𝑛)=𝜇(𝑊𝑛,𝑝−𝑣𝑛) by Lemma 2.12. Moreover, since 𝑊𝑛,𝑝−𝑣𝑛−1𝑣𝑛−𝑣1𝑣𝑝+𝑣𝑝𝑣𝑛+𝑣𝑛𝑣1 is the graph obtained from 𝑊𝑛,𝑝−𝑣𝑛 by subdividing the edge 𝑣1𝑣𝑝. Thus, by Lemma 2.11 it follows that 𝜇𝑊𝑛,𝑝−𝑣𝑛−1𝑣𝑛𝑊=𝜇𝑛,𝑝−𝑣𝑛𝑊>𝜇𝑛,𝑝−𝑣𝑛−1𝑣𝑛−𝑣1𝑣𝑝+𝑣𝑝𝑣𝑛+𝑣𝑛𝑣1𝑊=𝜇𝑛,𝑝+1.(2.9)
Therefore, 𝜇(𝑊𝑛,𝑝)>𝜇(𝑊𝑛,𝑝+1). Repeating the above process, we can conclude that 𝜇(𝑊𝑛,𝑝)>𝜇(𝑊𝑛,𝑝+1)>⋯>𝜇(𝑊𝑛,𝑛−1) holds for 𝑛−𝑝≥2.
By combining the above arguments, 𝑈≅𝑊𝑛,𝑛−1.

3. The Signless Laplacian Spreads of Unicyclic Graphs

In [1], the largest signless Laplacian spread among all unicyclic graphs with 𝑛 vrtices was determined, as follows.

Theorem 3.1 (see [1]). If 𝑛≥8 and 𝑈∈𝕌𝑛⧵{𝑆3𝑛}, then 𝑆𝑄(𝑆3𝑛)>𝑆𝑄(𝑈).

The next result extends the order of Theorem 3.1 to the first four largest values.

Theorem 3.2. If 𝑛≥16 and 𝑈∈𝕌𝑛⧵{𝑆3𝑛,𝑆4𝑛,𝐹𝑛,𝐻𝑛}, then 𝑆𝑆𝑄3𝑛𝑆>𝑆𝑄4𝑛𝐹>max𝑆𝑄𝑛𝐻,𝑆𝑄𝑛𝐹≥min𝑆𝑄𝑛𝐻,𝑆𝑄𝑛>𝑆𝑄(𝑈).(3.1)

Remark 3.3. With the aid of computer, we always have 𝑆𝑄(𝐹𝑛)<𝑆𝑄(𝐻𝑛). But it seems rather difficult to be proved.

To prove Theorem 3.2, we need to introduce more lemmas as follows.

Proposition 3.4. Suppose 𝑈 is a unicyclic graph on 𝑛 vertices with Δ≤𝑛−3. If 𝑛≥9, then 𝑆𝑄(𝑈)≤𝑛−1.1.

Proof. Note that 𝜇𝑛(𝑈)≥0 and 𝑆𝑄(𝑈)=𝜇1(𝑈)−𝜇𝑛(𝑈)≤𝜇1(𝑈). We only need to prove max{𝑑(𝑣)+𝑚(𝑣)∶𝑣∈𝑉}≤𝑛−1.1 by Lemma 2.3. Suppose 𝑑(𝑢)+𝑚(𝑢)=max{𝑑(𝑣)+𝑚(𝑣)∶𝑣∈𝑉}. We consider the next three cases.
Case 1 (𝑑(𝑢)=1). Suppose 𝑣∈𝑁(𝑢). Then, 𝑑(𝑢)+𝑚(𝑢)=1+𝑑(𝑣)≤1+Δ≤𝑛−2<𝑛−1.1.
Case 2 (𝑑(𝑢)=2). Suppose 𝑁(𝑢)={𝑤,𝑣}. Note that 𝑈 is a unicyclic graph. Then, |𝑁(𝑣)∩𝑁(𝑤)|≤2 and |𝑁(𝑣)∪𝑁(𝑤)|≤𝑛. Therefore, 𝑑(𝑢)+𝑚(𝑢)=2+𝑑(𝑣)+𝑑(𝑤)2≤2+𝑛+22<𝑛−1.1.(3.2)
Case 3 (3≤𝑑(𝑢)≤𝑛−3). Note that 𝑈 has 𝑛 edges and 3≤𝑑(𝑢)≤𝑛−3. By inequality (2.2), we have 𝑑(𝑢)+𝑚(𝑢)≤𝑑(𝑢)−1+2𝑛−2𝑑(𝑢).(3.3) Next we will prove that 𝑑(𝑢)−1+((2𝑛−2)/𝑑(𝑢))≤𝑛−1.1, equivalently, 𝑑(𝑢)(𝑛−𝑑(𝑢)−0.1)≥2𝑛−2. Let 𝑔(𝑥)=(𝑛−𝑥−0.1)𝑥, where 3≤𝑥≤𝑛−3. Since 𝑔′(𝑥)=𝑛−0.1−2𝑥 and 3≤𝑥≤𝑛−3, we have 𝑔(𝑥)≥min{𝑔(3),𝑔(𝑛−3)}>2𝑛−2.
By combining the above arguments, the result follows.

Lemma 3.5. If 𝑛≥16, then 𝑛−1.1<𝑆𝑄(𝐹𝑛)<𝑛−1.

Proof. Let 𝑓1(𝑥)=𝑥5−(𝑛+5)𝑥4+(6𝑛+3)𝑥3−(9𝑛−1)𝑥2+(3𝑛+8)𝑥−4. Clearly, 𝑓1112𝑛=−32𝑛580𝑛5−56𝑛4−32𝑛3−10𝑛2.+10𝑛−1(3.4) Next we will prove that 𝑓1(1/2𝑛)<0 when 𝑛≥16. Let 𝜓(𝑛)=80𝑛5−56𝑛4−32𝑛3−10𝑛2+10𝑛−1. When 𝑛≥16, since ğœ“î…žî…žî…ž(𝑛)=4800𝑛2−1344𝑛−192>0, we have ğœ“î…žî…ž(𝑛)=1600𝑛3−672𝑛2−192𝑛−20â‰¥ğœ“î…žî…ž(16)=6378476>0, then ğœ“î…ž(𝑛)=400𝑛4−224𝑛3−96𝑛2−20𝑛+10â‰¥ğœ“î…ž(16)=25272010>0. Thus, 𝜓(𝑛)≥𝜓(16)=80082591>0, and hence 𝑓1(1/2𝑛)<0.
With the similar method, we have 𝑓1(0.1)=0.2159𝑛−3.18749>0,𝑓11(0.5)=𝑓32(11−2𝑛)<0,1(3)=9𝑛−52>0,𝑓1(𝑛−1)=−20+21𝑛−5𝑛2𝑓<0,11𝑛−1+=12𝑛32𝑛516𝑛8−320𝑛7+1232𝑛6−1728𝑛5+1192𝑛4−504𝑛3+140𝑛2−20𝑛+1>0.(3.5)
By (2.5), we can conclude that 1/2𝑛<𝜇𝑛(𝐹𝑛)<0.1 and 𝑛−1<𝜇1(𝐹𝑛)<𝑛−1+1/2𝑛. Thus, 𝑛−1.1<𝑆𝑄(𝐹𝑛)=𝜇1(𝐹𝑛)−𝜇𝑛(𝐹𝑛)<𝑛−1.

Lemma 3.6. If 𝑛≥16, then 𝑛−1.1<𝑆𝑄(𝐻𝑛)<𝑛−1.

Proof. Let 𝑓2(𝑥)=𝑥5−(𝑛+5)𝑥4+(6𝑛+4)𝑥3−(10𝑛−2)𝑥2+(3𝑛+12)𝑥−4. It is easily checked that 𝑓2112𝑛=−32𝑛580𝑛5−112𝑛4−40𝑛3−14𝑛2𝑓+10𝑛−1<0,2(0.1)=0.2059𝑛−2.77649>0,𝑓21(0.5)=𝑓32(87−10𝑛)<0,2(2.8)>0.2464𝑛−2.14>0,𝑓2(𝑛−1)=−24+25𝑛−5𝑛2𝑓<0,21𝑛−1+=12𝑛32𝑛516𝑛8−320𝑛7+1376𝑛6−1888𝑛5+1288𝑛4−520𝑛3+144𝑛2−20𝑛+1>0.(3.6) By (2.6), we can conclude that 1/2𝑛<𝜇𝑛(𝐻𝑛)<0.1 and 𝑛−1<𝜇1(𝐻𝑛)<𝑛−1+(1/2𝑛). Thus, 𝑛−1.1<𝑆𝑄(𝐻𝑛)=𝜇1(𝐻𝑛)−𝜇𝑛(𝐻𝑛)<𝑛−1.

Proof of Theorem 3.2. By Lemma 2.6 and (2.7), 𝑆𝑄(𝑆4𝑛)=𝜇1(𝑆4𝑛)−𝜇𝑛(𝑆4𝑛)=𝜇1(𝑆4𝑛)>𝑛−1. Note that 𝑆3𝑛 is the unique unicyclic graph with Δ=𝑛−1, and 𝐹𝑛, 𝐻𝑛, 𝑆4𝑛 are all the unicyclic graphs with Δ=𝑛−2. Now suppose 𝑈∈𝕌𝑛⧵{𝑆3𝑛,𝐹𝑛,𝐻𝑛,𝑆4𝑛}. Then, Δ(𝑈)≤𝑛−3. By Lemmas 3.5 and 3.6, Theorem 3.1, and Proposition 3.4, we can conclude that 𝑆𝑆𝑄3𝑛𝑆>𝑆𝑄4𝑛𝐹>𝑛−1>max𝑆𝑄𝑛𝐻,𝑆𝑄𝑛𝐹≥min𝑆𝑄𝑛𝐻,𝑆𝑄𝑛>𝑛−1.1≥𝑆𝑄(𝑈).(3.7) This completes the proof of Theorem 3.2.

4. A Class of Unicyclic Graphs Determined by Their Signless Laplacian Spectra

A graph 𝐺 is said to be determined by its signless Laplacian spectrum if there does not exist other nonisomorphic graph 𝐻 such that 𝐻 and 𝐺 share the same signless Laplacian spectra (see [11]). Let 𝑆3(𝑛,𝑘) be the unicyclic graph on 𝑛 vertices obtained by attaching 𝑘, and 𝑛−𝑘−3 pendant vetrices to two vertices of 𝐶3, respectively. By the definition, 𝑆3(𝑛,𝑛−3)=𝑆3𝑛. The next theorem is the main result of this section.

Theorem 4.1. For any 𝑘≥⌈𝑛/2⌉−1, if 8𝑘(𝑛−3−𝑘)≠9(𝑛−3), then 𝑆3(𝑛,𝑘) is determined by its signless Laplacian spectrum.

To prove Theorem 4.1, we need some more lemmas as follows.

Lemma 4.2 (see [12]). If 𝐺 is a graph on 𝑛 vertices with vertex degrees 𝑑1≥𝑑2≥⋯≥𝑑𝑛 and signless Laplacian eigenvalues 𝜇1≥𝜇2≥⋯≥𝜇𝑛, then 𝜇2≥𝑑2−1. Moreover, if 𝜇2=𝑑2−1, then 𝑑1=𝑑2, and the maximum and the second maximum degree vertices are adjacent.

Lemma 4.3 (see [12]). If 𝐺 is a connected graph with 𝑛 vertices, then 𝜇𝑛(𝐺)<𝛿(𝐺).

Lemma 4.4 (see [7]). In any graph, the multiplicity of the eigenvalue 0 of the signless Laplacian matrix of 𝐺 is equal to the number of bipartite components of 𝐺.

Let 𝕌(𝑛,Δ) be the class of unicyclic graphs on 𝑛 vertices with maximum degree Δ.

Lemma 4.5 (see [13]). For any 𝑘≥⌈𝑛/2⌉−1, if 𝑈∈𝕌(𝑛,𝑘+2), then 𝜇(𝑈)≤𝜇(𝑆3(𝑛,𝑘)), where the equality holds if and only if 𝑈≅𝑆3(𝑛,𝑘).

Lemma 4.6 (see [13]). Let 𝐺 be the graph with the largest signless Laplacian spectral radius in 𝕌(𝑛,Δ). If Δ≤𝑛−2, then there must exist some graph 𝐺1∈𝕌(𝑛,Δ+1) such that 𝜇(𝐺)<𝜇(𝐺1).

Proof of Theorem 4.1. By an elementary computation, we have Φ𝑆3(𝑛,𝑘),𝑥=(𝑥−1)𝑛−5𝑓3(𝑥),(4.1) where 𝑓3(𝑥)=𝑥5−(𝑛+5)𝑥4+(𝑘𝑛+5𝑛−𝑘2−3𝑘+7)𝑥3−(2𝑘𝑛+7𝑛−2𝑘2−6𝑘+7)𝑥2+(3𝑛+8)𝑥−4. Now suppose that there exists another graph 𝐺 such that 𝐺 and 𝑆3(𝑛,𝑘) share the same signless Laplacian spectra. Next we will prove that 𝐺≅𝑆3(𝑛,𝑘). We only need to prove the following facts.
Fact 1. 𝐺 is a connected unicyclic graph.Proof of Fact 1. Assume that 𝐺 has exactly 𝑡 connected components, say 𝐺1,…,𝐺𝑡, where 𝑡≥1. By Lemma 4.4, 𝐺𝑖 is not a bipartite graph for 1≤𝑖≤𝑡 because 𝑆3(𝑛,𝑘) is not a biparite graph. Thus, 𝐺𝑖 is a connected unicyclic graph for 1≤𝑖≤𝑡 because 𝐺 has 𝑛 edges (since 𝑆3(𝑛,𝑘) has 𝑛 edges). Moreover, since 8𝑘(𝑛−3−𝑘)≠9(𝑛−3), we have 𝑓3(4)≠0. Thus, 𝐺𝑖 is not a cycle for 1≤𝑖≤𝑡 because 4 is the eigenvalue of the signless Laplacian matrix of a cycle. By the a bove arguments, we can conclude that 𝐺𝑖 is not a bipartite graph and has at least one pendant vertex. Thus, 𝐺𝑖 has at least two signless Laplacian eigenvalues being larger than 1 by Lemma 4.2, and the smallest signless Laplacian eigenvalue of 𝐺𝑖 is less than 1 by Lemma 4.3. Therefore, by (4.1) we can conclude that 𝑡≤1, and hence 𝐺 is connected. Clearly, 𝐺 is a connected unicyclic graph because 𝐺 has 𝑛 edges.Fact 2. Δ(𝐺)≤𝑘+2. Proof of Fact 2. Note that 𝑘≥⌈𝑛/2⌉−1. Then, 𝑛<2𝑘+3. By Lemmas 2.3 and 2.6, 𝑆Δ(𝐺)+1≤𝜇(𝐺)=𝜇3(𝑛,𝑘)≤max𝑘+2+𝑛+1𝑘+2,𝑛−1−𝑘+𝑛+1𝑛−𝑘−1,2+𝑛+12<𝑘+4.(4.2) Thus, Δ(𝐺)≤𝑘+2 holds.Fact 3. 𝐺≅𝑆3(𝑛,𝑘). Proof of Fact 3. By Fact 2, we have Δ(𝐺)≤𝑘+2. If Δ(𝐺)≤𝑘+1<Δ(𝑆3(𝑛,𝑘)), then 𝜇(𝐺)<𝜇(𝑆3(𝑛,𝑘)) by Lemmas 4.5 and 4.6, a contradiction. Thus, Δ(𝐺)=𝑘+2, and hence the result follows from Lemma 4.5 because 𝜇(𝐺)=𝜇(𝑆3(𝑛,𝑘)).
This completes the proof of Theorem 4.1.

Acknowledgments

This work is supported by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. LYM10039) and NNSF of China (no. 11071088).

References

  1. M. H. Liu and B. L. Liu, “The signless Laplacian spread,” Linear Algebra and Its Applications, vol. 432, no. 2-3, pp. 505–514, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. C. S. Oliveira, L. S. de Lima, N. M. M. de Abreu, and S. Kirkland, “Bounds on the Q-spread of a graph,” Linear Algebra and Its Applications, vol. 432, no. 9, pp. 2342–2351, 2010. View at: Publisher Site | Google Scholar
  3. D. Cvetković and S. K. Simić, “Towards a spectral theory of graphs based on the signless Laplacian. I,” Institut Mathématique Publications, vol. 85(99), pp. 19–33, 2009. View at: Publisher Site | Google Scholar
  4. D. Cvetković and S. K. Simić, “Towards a spectral theory of graphs based on the signless Laplacian. II,” Linear Algebra and Its Applications, vol. 432, no. 9, pp. 2257–2272, 2010. View at: Publisher Site | Google Scholar
  5. D. Cvetković and S. K. Simić, “Towards a spectral theory of graphs based on the signless Laplacian. III,” Applicable Analysis and Discrete Mathematics, vol. 4, no. 1, pp. 156–166, 2010. View at: Publisher Site | Google Scholar
  6. D. Cvetković, P. Rowlinson, and S. K. Simić, “Eigenvalue bounds for the signless Laplacian,” Institut Mathématique Publications, vol. 81(95), pp. 11–27, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. D. Cvetković, P. Rowlinson, and S. K. Simić, “Signless Laplacians of finite graphs,” Linear Algebra and Its Applications, vol. 423, no. 1, pp. 155–171, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. K. C. Das, “The Laplacian spectrum of a graph,” Computers & Mathematics with Applications, vol. 48, no. 5-6, pp. 715–724, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. M. H. Liu and B. L. Liu, “The method to calculate the characteristic polynomial of graphs on n vertices by the aid of computer” (Chinese), Submitted to Numerical Mathematics Journal. View at: Google Scholar
  10. M. H. Liu, X. Z. Tan, and B. L. Liu, “The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices,” Czechoslovak Mathematical Journal, vol. 60, no. 3, pp. 849–867, 2010. View at: Publisher Site | Google Scholar
  11. E. R. van Dam and W. H. Haemers, “Which graphs are determined by their spectrum?” Linear Algebra and Its Applications, vol. 373, pp. 241–272, 2003, Special issue on the Combinatorial Matrix Theory Conference. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. K. C. Das, “On conjectures involving second largest signless Laplacian eigenvalue of graphs,” Linear Algebra and Its Applications, vol. 432, no. 11, pp. 3018–3029, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. M. H. Liu and B. L. Liu, “On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree,” Linear Algebra and Its Applications, vol. 435, pp. 3045–3055, 2011. View at: Google Scholar

Copyright © 2011 Muhuo Liu. 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.

696 Views | 662 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder