Research Article | Open Access

# On the Signless Laplacian Spectral Radius of Bicyclic Graphs with Perfect Matchings

**Academic Editor:**Hung Nguyen-Xuan

#### Abstract

The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.

#### 1. Introduction

Let be a simple connected graph with vertex set and edge set . Its adjacency matrix is defined as matrix , where if is adjacent to , and , otherwise. Denote by or the degree of the vertex . Let be the signless Laplacian matrix of graph , where denotes the diagonal matrix of vertex degrees of . It is well known that is a real symmetric matrix and is a positive semidefinite matrix. The largest eigenvalues of and are called the spectral radius and the signless Laplacian spectral radius of , denoted by and , respectively. When is connected, and are a nonnegative irreducible matrix. By the well-known Perron-Frobenius theory, is simple and has a unique positive unit eigenvector and so does . We refer to such an eigenvector corresponding to as the Perron vector of .

Two distinct edges in a graph are independent if they are not adjacent in . A set of mutually independent edges of is called a matching of . A matching of maximum cardinality is a maximum matching in . A matching that satisfies is called a perfect matching of the graph . Denote by and the cycle and the path on vertices, respectively.

The characteristic polynomial of is , which is denoted by or . The characteristic polynomial of is , which is denoted by or .

A bicyclic graph is a connected graph in which the number of vertices equals the number of edges minus one. Let and be two vertex-disjoint cycles. Suppose that is a vertex of and is a vertex of . Joining and by a path of length , where and means identifying with , denoted by , is called an graph (see Figure 1). Let , and be the three vertex-disjoint paths, where , and at most one of them is 1. Identifying the three initial vertices and the three terminal vertices of them, respectively, denoted by , is called a graph (see Figure 2).

Let be the set of all bicyclic graphs on vertices with perfect matchings. Obviously consists of two types of graphs: one type, denoted by , is a set of graphs each of which is an graph with trees attached; the other type, denoted by , is a set of graphs each of which is graph with trees attached. Then we have .

The investigation on the spectral radius of graphs is an important topic in the theory of graph spectra, in which some early results can go back to the very beginnings (see [1]). The recent developments on this topic also involve the problem concerning graphs with maximal or minimal spectral radius of a given class of graphs. In [2], Chang and Tian gave the first two spectral radii of unicyclic graphs with perfect matchings. Recently, Yu and Tian [3] gave the first two spectral radii of unicyclic graphs with a given matching number; Guo [4] gave the first six spectral radii over the class of unicyclic graphs on a given number of vertices; and Guo [5] gave the first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings. For more results on this topic, the reader is referred to [6–9] and the references therein.

In this paper, we deal with the extremal signless Laplacian spectral radius problems for the bicyclic graphs with perfect matchings. The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.

#### 2. Lemmas

Let or denote the graph obtained from by deleting the vertex or the edge . A pendant vertex of is a vertex with degree . A path in is called a pendant path if and . If , then we say is a pendant edge of the graph .

In order to complete the proof of our main result, we need the following lemmas.

Lemma 1 (see [10, 11]). *Let be a connected graph and two vertices of . Suppose that and is the Perron vector of , where corresponds to the vertex . Let be the graph obtained from by deleting the edges and adding the edges . If , then .*

The cardinality of a maximum matching of is commonly known as its matching number, denoted by .

From Lemma 1, we have the following results.

Corollary 2. *Let and be two vertices in a connected graph and suppose that paths of length are attached to at and paths of length are attached to at to form . Then either or or .*

Corollary 3. *Suppose is a vertex of graph with . Let be a graph obtained by attaching a pendant edge to at . Suppose paths of length are attached to at to form . Let
**
If has a perfect matching, then we have that has a perfect matching and
*

An internal path of a graph is a sequence of vertices with such that(1)the vertices in the sequences are distinct (except possibly );(2) is adjacent to ;(3)the vertex degrees satisfy , (unless ) and .

Let be a connected graph, and . The graph is obtained from by subdividing the edge , that is, adding a new vertex and edges in . By similar reasoning as that of Theorem 3.1 of [12], we have the following result.

Lemma 4. *Let be an internal path of a connected graph . Let be a graph obtained from by subdividing some edge of . Then we have .*

Corollary 5. *Suppose that is an internal path of the graph and for . Let be the graph obtained from by amalgamating , , and to form a new vertex together with attaching a new pendant path of length at . Then and .*

*Proof. *From Lemma 4 and the well-known Perron-Frobenius theorem, It is easy to prove that . Next, we prove that . Let be a maximum matching of . If or , then or is a matching of . Thus, ; If and , then there exist two edges and . Thus, is a matching of . Hence, , completing the proof.

Let be the subdivision graph of obtained by subdividing every edge of .

Lemma 6 (see [13, 14]). *Let be a graph on vertices and edges, , . Then .*

Lemma 7 (see [15]). *Let be a vertex of a connected graph . Let be the graph obtained from by attaching two pendant paths of lengths and at , respectively. If , then .*

Corollary 8. *Suppose that is a pendant path of the graph with . Let be the graph obtained from by amalgamating , , and to form a new vertex together with attaching a new pendant path of length at . Then and .*

*Proof. *By Lemma 7 we have . By the proof as that of Corollary 5, we have .

Lemma 9 (see [16]). *Let be an edge of , and let be the set of all circuits containing . Then satisfies
**
where the summation extends over all .*

Lemma 10 (see [16]). *Let be a vertex of , and let be the collection of circuits containing , and let denote the set of vertices in the circuit . Then the characteristic polynomial satisfies
**
where the first summation extends over those vertices adjacent to , and the second summation extends over all .*

Lemma 11 (see [17]). *Let be a connected graph, and let be a proper spanning subgraph of . Then , and, for .*

Let denote the maximum degree of . From Lemma 11, we have .

Lemma 12 (see [13]). *Let be a connected graph, and let be a proper spanning subgraph of . Then .*

Lemma 13 (see [18]). *Let be a connected graph with vertex set . Suppose that , , , , , , and . Let be the graph obtained from by amalgamating and to form a new vertex together with subdivising the edge with a new vertex . If , then*(1)

*either*(2)

*or*;

*and*.Lemma 14 (see [18]). *Suppose is a vertex of the bicyclic graph with . Let be a graph obtained by attaching a pendant edge to at . Suppose that a pendant edge and paths of length are attached to at to form . Let . Then we have*(1)*, ;*(2)*.*

#### 3. Main Results

Lemma 15. *Let be the graphs as Figure 3. Then for , we have , .*

*Proof. *From Lemma 10, we have
From (5), we have
If , for , it is easy to prove that . Hence, for . When , by direct calculation, we also get , respectively. So, for . By Lemma 6, we know that . Hence, . By similar method, the result is as follows.

Theorem 16. *If , then , with equality if and only if .*

*Proof. *Let be the Perron vector of . From Lemma 12 and by direct calculations, we have, for , . So, in the following, we only consider those graphs, which have signless Laplacian spectral radius greater than .

Choose such that is as large as possible. Then consists of a subgraph which is one of graphs , , and (see Figures 1 and 2).

Let be a tree attached at some vertex, say, , of ; is the number of vertices of including the vertex . In the following, we prove that tree is formed by attaching at most one path of length 1 and other paths of length 2 at .

Suppose is a pendant path of and is a pendant vertex. If , let . From Corollary 8, we have and , which is a contradiction.

For each vertex , we prove that . Otherwise, there must exist some vertex of such that . From the above proof, we have the pendant paths attached which have length of at most . Obviously, there exists an internal path between and some vertex of , denoted by . If , let be the graph obtained from by amalgamating , , and to form a new vertex together with attaching a new pendant path of length 2 at . From Corollary 5, we have and , which is a contradiction. If , by Lemma 14 and Corollary 3, we can get a new graph such that and , which is a contradiction.

From the proof as above, we have the tree which is obtained by attaching some pendant paths of length and at most one pendant path of length at .

From Corollary 2, we have all the pendant paths of length in which must be attached at the same vertex of .

In the following, we prove that is isomorphic to one of graphs (see Figure 3). We distinguish the following two cases:*Case 1 (**)*. We prove that is isomorphic to one of graphs , , and .

Assume that there exists some cycle of with length of at least 4. From Corollary 5, we have each internal path of , which is not a triangle, has length . Note that all the pendant paths of length in must be attached at the same vertex, then there must exist edges , , and and , , , and . Let be the graph obtained from by amalgamating and to form a new vertex together with subdividing the edge with a new vertex . From Lemma 13, we have and either or , which is a contradiction. Then for each cycle of , we have .

Assume that . If there exists an internal path with length greater than in . Then, by Corollary 5, we can get a new graph such that and , which is a contradiction. Thus, and either or . By Lemma 13, we can also get a new graph such that and , which is a contradiction. Hence, .

We distinguish the following three subcases:*Subcase 1.1 (**)*. Then is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of , where is one of graphs (see Figure 4).

Assume that . If , let ; if , let . Obviously, and either or by Lemma 1, which is a contradiction. By similar reasoning, we have also .*Subcase 1.2 (**)*. Then is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of , where is one of graphs (see Figure 4).

Assume that . If , let ; if , let . Obviously, and either or by Lemma 1, which is a contradiction. By similar reasoning, we have also .*Subcase 1.3 (**)*. Then is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of , where is one of graphs (see Figure 4).

Assume that . If , let ; if , let . Obviously, and either or by Lemma 1, a contradiction. By similar reasoning, we have also .

Thus, is isomorphic to one of the graphs , and . In the following, we prove that is isomorphic to one of graphs , and .

Assume that is obtained by attaching all the pendant paths of length 2 at vertex of . If , let be the graph obtained from by attaching pendant paths of length 2 at . If , let . Obviously, and by Lemma 1, a contradiction. Then . By similar reasoning, the result follows.*Case 2 (**)*. By similar reasoning as that of Case 1, we have is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of , where is one of graphs (see Figure 4).

From Lemma 1, it is easy to prove that and all the pendant paths of length 2 are attached at the vertex of degree 3 of or of degree 4 of . Thus, is isomorphic to one of graphs , and (see Figure 3).

So, is isomorphic to one of graphs . From Lemma 15, we know . Thus, .

#### Conflict of Interests

The authors declare that they have no competing interests.

#### Authors’ Contribution

All authors completed the paper together. All authors read and approved the final paper.

#### Acknowledgments

This research is supported by NSFC (nos. 10871204, 61370147, and 61170309) and by Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020).

#### References

- L. Collatz and U. Sinogowitz, “Spektren endlicher grafen,”
*Mathematisches Seminar der Universität Hamburg*, vol. 21, no. 1, pp. 63–77, 1957. View at: Publisher Site | Google Scholar - A. Chang and F. Tian, “On the spectral radius of unicyclic graphs with perfect matchings,”
*Linear Algebra and Its Applications*, vol. 370, pp. 237–250, 2003. View at: Publisher Site | Google Scholar - A. Yu and F. Tian, “On the spectral radius of unicyclic graphs,”
*MATCH Communications in Mathematical and in Computer Chemistry*, no. 51, pp. 97–109, 2004. View at: Google Scholar - S. G. Guo, “First six unicyclic graphs of order
*n*with largest spectral radius,”*Applied Mathematics Journal of Chinese Universities A*, vol. 18, no. 4, pp. 480–486, 2003. View at: Google Scholar - J.-M. Guo, “On the spectral radii of unicyclic graphs with fixed matching number,”
*Discrete Mathematics*, vol. 308, no. 24, pp. 6115–6131, 2008. View at: Publisher Site | Google Scholar - R. A. Brualdi and E. S. Solheid, “On the spectral radius of connected graphs,”
*Publications de l'Institut Mathématique*, vol. 39, no. 53, pp. 45–54, 1986. View at: Google Scholar - D. Cvetković and P. Rowlinson, “Spectra of unicyclic graphs,”
*Graphs and Combinatorics*, vol. 3, no. 1, pp. 7–23, 1987. View at: Publisher Site | Google Scholar - Y. Hong, “On the spectra of unicycle graph,”
*East China Normal University. Natural Science Edition*, vol. 1, pp. 31–34, 1986. View at: Google Scholar - A. Chang, F. Tian, and A. Yu, “On the index of bicyclic graphs with perfect matchings,”
*Discrete Mathematics*, vol. 283, no. 1–3, pp. 51–59, 2004. View at: Publisher Site | Google Scholar - 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 - Y. Hong and X. D. Zhang, “Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees,”
*Discrete Mathematics*, vol. 296, no. 2-3, pp. 187–197, 2005. View at: Publisher Site | Google Scholar - J.-M. Guo, “The Laplacian spectral radius of a graph under perturbation,”
*Computers and Mathematics with Applications*, vol. 54, no. 5, pp. 709–720, 2007. View at: Publisher Site | Google Scholar - D. Cvetković, M. Doob, and H. Sachs,
*Spectra of Graphs*, Johann Ambrosius Barth, Heidelberg, Germany, 3rd edition, 1995. - B. Zhou and I. Gutman, “A connection between ordinary and Laplacian spectra of bipartite graphs,”
*Linear and Multilinear Algebra*, vol. 56, no. 3, pp. 305–310, 2008. View at: Publisher Site | Google Scholar - D. Cvetković and S. K. Simić, “Towards a spectral theory of graphs based on the signless Laplacian I,”
*Publications de l'Institut Mathematique*, vol. 85, no. 99, pp. 19–33, 2009. View at: Google Scholar - A. J. Schwenk and R. J. Wilson, “On the eigenvalues of a graph,” in
*Selected Topics in Graph Theory*, L. W. Beineke and R. J. Wilson, Eds., chapter 2, pp. 307–336, Academic Press, London, UK, 1978. View at: Google Scholar - Q. Li and K. Q. Feng, “On the largest eigenvalue of graphs,”
*Acta Mathematicae Applicatae Sinica*, vol. 2, pp. 167–175, 1979 (Chinese). View at: Google Scholar - J.-M. Zhang, T.-Z. Huang, and J.-M. Guo, “On the signless Laplacian spectral radius of unicyclic graphs with fixed matching number,” In press. View at: Google Scholar

#### Copyright

Copyright © 2014 Jing-Ming Zhang et al. 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.