#### 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, is an undirected simple graph with vertices and edges. If is connected with , then is called a -cyclic graph. Especially, if or 1, then is called a tree or a unicyclic graph, respectively. Let be the class of unicyclic graphs with vertices. The neighbor set of a vertex is denoted by . We write for the degree of vertex . In particular, let and be the maximum degree and minimum degree of , respectively. Let be the adjacency matrix and be the diagonal matrix whose -entry is , of , respectively. The signless Laplacian matrix of is . Clearly, is positive semidefinite  and its eigenvalues can be arranged as Let be the signless Laplacian spectral radius of , namely, . The signless Laplacian spread of is defined as [1, 2]. Let be the signless Laplacian characteristic polynomial of , that is, .

Recently, the research on the spectrum of receive much attention. Some properties of signless Laplacian spectra of graphs and some possibilities for developing the spectral theory of graphs based on are discussed in . The largest signless Laplacian spectral radius and the largest signless Laplacian spread among the class of unicyclic graphs with vertices were determined in  and , respectively. The smallest signless Laplacian spectral radius among the class of unicyclic graphs with vertices was determined in . 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 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 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 , , and be the star, path, and cycle with vertices, respectively. In the following, let be the unicyclic graph obtained by adding one edge to two pendant vertices of , and let , , be the unicyclic graphs with vertices as shown in Figure 1.

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

Theorem 2.1 (see ). If , and , then , where the equality holds if and only if .

Theorem 2.2. Suppose , and . (1) If , then , where the equality holds if and only if , and equals the maximum root of the equation . (2) If and , then , where the equality holds if and only if , and equals the maximum root of the equation . (3) If , and , then , where the equality holds if and only if , and equals the maximum root of the equation .

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

Lemma 2.3 (see ). , where .

Proposition 2.4. Suppose and is a -cyclic graph on vertices with . If , then .

Proof. We only need to prove that by Lemma 2.3. Suppose . We consider the next three cases.
Case 1 (). Suppose . Then, .
Case 2 (). Suppose that . Then,
Case 3 (). Note that has edges and . Then, Next we will prove that , equivalently, . Let , where . Since and , we have .
By combining the above arguments, the result follows.

Corollary 2.5. Suppose . If and , then .

Lemma 2.6 (see ). If is a connected graph of order , then , where the equality holds if and only if .

Suppose is a square matrix, let be the entry appearing in the th row and the th column of . The next result gives a new method to calculate the signless Lapalacian characteristic polynomial of an -vertex graph via the aid of computer.

Lemma 2.7 (see ). Let be a graph on () vertices with . If is obtained from by attaching new pendant vertices, say , to , then where is corresponding to the vertex , and .

Example 2.8. Let be the unicyclic graph as shown in Figure 1. By Lemma 2.7, we have By using “Matlab”, it easily follows that With the similar method, by Lemma 2.7 we have

Proof of Theorem 2.2. Note that is the unique unicyclic graph with , and , , are all the unicyclic graphs with . Now suppose . By Lemma 2.6 and Corollary 2.5, we have because .
To finish the proof of Theorem 2.2, we only need to show that by Theorem 2.1. By Lemma 2.6, it follows that , and . When and , by (2.5), (2.6) and (2.7), it follows that Therefore, we have . Thus, Theorem 2.2 follows.

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

Theorem 2.9 (see ). If , then , where the equality holds if and only if .

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

Theorem 2.10. For any , if , then .

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

Let be a connected graph, and . The graph is obtained from by subdividing the edge , that is, adding a new vertex and edges in . An internal path, say , is a path joining and (which need not be distinct) such that and have degree greater than 2, while all other vertices are of degree 2.

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

By , we mean that is a subgraph of and .

Lemma 2.12 (see ). If and is a connected graph, then .

If and , then we called a branching point of . Let be a connected unicyclic graph and be a tree such that is attached to a vertex of the unique cycle of . The vertex is called the root of , and is called a root tree of . Throughout this paper, we assume that does not include the root . Clearly, is obtained by attaching root trees to some vertices of the unique cycle of .

Proof of Theorem 2.10. In the proof of this result, we assume that the unique cycle of is . Now choose such that is as small as possible. Since , has at least one branching point. We consider the next two cases.
Case 1. There are at least two branching points in .
Now suppose and are two branching points in such that there does not exsit other barnching point between the shortest path in connected and . Let be the shortest path in connected and . Since is a branching point of , there is at least one pendant vertex, say , in . Suppose is the unique neighbor vertex of in (may be ). Let and . Then, , and hence by Lemma 2.12. Let . By the hypothesis, is an internal path of . By Lemma 2.11, we can conclude that . But is also a unicyclic graph with vertices and because is a branching point of , it is a contradiction to the choice of . Thus, Case 1 is impossible.
Case 2. There is unique branching point in .
Subcase 1. There is at least a branching point outside .
It can be proved analogously with Case 1.

Subcase 2. There does not exist any branching point outside .Suppose is the unique brancing poing in . By the hypothesis, is also the unique branching point of . Then, is obtained by attaching paths to the vertex of .
If , then there are at least two paths being attaching to . It can be proved analogously with Case 1.
If , then is a lollipop graph, that is, . Let and ,. If , since , by Lemma 2.12. Moreover, since is the graph obtained from by subdividing the edge . Thus, by Lemma 2.11 it follows that
Therefore, . Repeating the above process, we can conclude that holds for .
By combining the above arguments, .

#### 3. The Signless Laplacian Spreads of Unicyclic Graphs

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

Theorem 3.1 (see ). If and , then .

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

Theorem 3.2. If and , then

Remark 3.3. With the aid of computer, we always have . 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 is a unicyclic graph on vertices with . If , then .

Proof. Note that and . We only need to prove by Lemma 2.3. Suppose . We consider the next three cases.
Case 1 (). Suppose . Then, .
Case 2 (). Suppose . Note that is a unicyclic graph. Then, and . Therefore,
Case 3 (). Note that has edges and . By inequality (2.2), we have Next we will prove that , equivalently, . Let , where . Since and , we have .
By combining the above arguments, the result follows.

Lemma 3.5. If , then .

Proof. Let . Clearly, Next we will prove that when . Let . When , since , we have , then . Thus, , and hence .
With the similar method, we have
By (2.5), we can conclude that and . Thus, .

Lemma 3.6. If , then .

Proof. Let . It is easily checked that By (2.6), we can conclude that and . Thus, .

Proof of Theorem 3.2. By Lemma 2.6 and (2.7), . Note that is the unique unicyclic graph with , and , , are all the unicyclic graphs with . Now suppose . Then, . By Lemmas 3.5 and 3.6, Theorem 3.1, and Proposition 3.4, we can conclude that This completes the proof of Theorem 3.2.

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

A graph is said to be determined by its signless Laplacian spectrum if there does not exist other nonisomorphic graph such that and share the same signless Laplacian spectra (see ). Let be the unicyclic graph on vertices obtained by attaching , and pendant vetrices to two vertices of , respectively. By the definition, . The next theorem is the main result of this section.

Theorem 4.1. For any , if , then is determined by its signless Laplacian spectrum.

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

Lemma 4.2 (see ). If is a graph on vertices with vertex degrees and signless Laplacian eigenvalues , then . Moreover, if , then , and the maximum and the second maximum degree vertices are adjacent.

Lemma 4.3 (see ). If is a connected graph with vertices, then .

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

Let be the class of unicyclic graphs on vertices with maximum degree .

Lemma 4.5 (see ). For any , if , then , where the equality holds if and only if .

Lemma 4.6 (see ). Let be the graph with the largest signless Laplacian spectral radius in . If , then there must exist some graph such that .

Proof of Theorem 4.1. By an elementary computation, we have where . Now suppose that there exists another graph such that and share the same signless Laplacian spectra. Next we will prove that . We only need to prove the following facts.
Fact 1. is a connected unicyclic graph.Proof of Fact 1. Assume that has exactly connected components, say , where . By Lemma 4.4, is not a bipartite graph for because is not a biparite graph. Thus, is a connected unicyclic graph for because has edges (since has edges). Moreover, since , we have . Thus, is not a cycle for because 4 is the eigenvalue of the signless Laplacian matrix of a cycle. By the a bove arguments, we can conclude that is not a bipartite graph and has at least one pendant vertex. Thus, has at least two signless Laplacian eigenvalues being larger than 1 by Lemma 4.2, and the smallest signless Laplacian eigenvalue of is less than 1 by Lemma 4.3. Therefore, by (4.1) we can conclude that , and hence is connected. Clearly, is a connected unicyclic graph because has edges.Fact 2. . Proof of Fact 2. Note that . Then, . By Lemmas 2.3 and 2.6, Thus, holds.Fact 3. . Proof of Fact 3. By Fact 2, we have . If , then by Lemmas 4.5 and 4.6, a contradiction. Thus, , and hence the result follows from Lemma 4.5 because .
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).