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Journal of Applied Mathematics
Volume 2013, Article ID 730396, 6 pages
http://dx.doi.org/10.1155/2013/730396
Research Article

The Laplacian Spectral Radius of a Class of Unicyclic Graphs

Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan 030024, China

Received 30 June 2013; Accepted 25 November 2013

Academic Editor: Ram N. Mohapatra

Copyright © 2013 Haixia 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.

Abstract

Let be the set of all unicyclic graphs with vertices and cycle length . For any , consists of the (unique) cycle (say ) of length and a certain number of trees attached to the vertices of having (in total) edges. If there are at most two trees attached to the vertices of , where is even, we identify in the class of unicyclic graphs those graphs whose Laplacian spectral radii are minimal.

1. Introduction

Following [1], let be a simple connected graph on vertices and edges (so is its order and is its size). For , the degree of is denoted by . Let be the maximum degree of . For two vertices and   () in , the distance between and , denoted by , is the number of edges in a shortest path joining and .

Let be the set of all unicyclic graphs on vertices and the cycle length . So, if , then consists of the (unique) cycle (say ) of length and a certain number of trees attached to vertices of having (in total) edges. If the cycle length of a unicyclic graph is even (odd), we call it an even (odd) unicyclic graph. We may assume that vertices of are or for short only (ordered in a natural way around , say, in the clockwise direction). For each , let be a rooted tree (with as its root) attached to . Then, for each , we can write . If , for each , is a path , whose root is a vertex of minimum degree, then we write .

Let be the adjacency matrix of and the diagonal matrix. Then the Laplacian matrix of is . Since is real symmetric and positive semidefinite, its eigenvalues are nonnegative real numbers. For a graph , we denote by the largest eigenvalue of and call it the Laplacian spectral radius.

The investigation on the Laplacian spectral radius of graphs is an important topic in the theory of graph spectra. Since 1980s, there are several monographs and a lot of research papers published continually (see [27]). Recently, the problem concerning graphs with maximal or minimal Laplacian spectral radius of a given class of graphs has been studied by many authors. Guo [8] determined the first four graphs with the largest Laplacian spectral radius among all unicyclic graphs on vertices.

In this paper, for any , if there are at most two trees attached to vertices of , where is even, we characterize in the class of even unicyclic graphs those graphs whose Laplacian spectral radii are minimal.

2. Main Results and Proofs

Let be the principal submatrix obtained from by deleting the corresponding row and column of . Generally, let be the principal submatrix obtained from by deleting the corresponding rows and columns of all vertices of . For any square matrix , denote by the characteristic polynomial of . In particular, if , we write by for convenience. If , then suppose that .

Lemma 1 (see [9]). Let and be two graphs. If for , then .

Lemma 2 (see [4]). Let be a graph containing at least one edge. Then . Moreover, if is connected on vertices, the equality holds if and only if .

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

Lemma 3 (see [10]). Let be a connected bipartite graph with vertex set and an edge on an internal path () of . Let () be the graph obtained from by subdividing the edge into new edges. Then one has .

Let be a vertex of a connected graph with at least two vertices. Let () be the graph obtained from by attaching two new paths and of length and , respectively, at , where and are distinct new vertices. Let .

Lemma 4 (see [11]). If is a connected bipartite graph on vertices and is a vertex of , let be the graph defined as above. If , then .

Lemma 5 (see [12]). Let be the matrix obtained from by deleting the rows and columns corresponding to two end vertices of , and suppose that , ; then(i);(ii);(iii), ;(iv).

From Lemma 5(i), all eigenvalues of are , where . Other characterizations of can be shown as follows.

Lemma 6 (see [9]). Let , where .

Lemma 7. For , .

Proof. It is easily obtained from some properties of the determinant .

In the following, we give a transformation property of unicyclic graphs.

Lemma 8. Let and be two nonadjacent vertices of the cycle , and suppose that ; that is, is the number of vertices in the shortest path , not including and , where . Let , where only and () are nonzero. If , then .

Proof. Case 1 (). Label vertices of properly; we have where and and are adjacent to and , respectively. is the column vector whose only nonzero entry is in the th position, and is the column vector whose only nonzero entry is in the first position.
By a direct computation, we can obtain where Furthermore, where is adjacent to . is the column vector, whose only nonzero entry is in the last position. is the column vector, whose only nonzero entry is in the first position.
Note that Similarly,
Combing the equations previous, we get where From Lemma 2, . By Lemmas 5(ii) and 6 and (7), for .
So holds from Lemma 1.
When , we appropriately modify and get the same result.

Next we consider some special even unicyclic graphs .

By Lemmas 6 and 7, we have From Lemmas 5(ii) and 5(iv) and (10), we get By (11) and of (10),

Case 1. The unicyclic graph , where (), is as follows.
Note that In view of (2) and (12)-(13), we have By Lemmas 5(ii) and 7, So,
Let and be the largest roots of and , respectively. In the proof of Theorem 2 (see [13]), we have shown that the sequence is strictly increasing by the initial value . For any fixed , the second largest root is less than 4.
By Lemmas 5(ii) and 7, we get where for by derivative.
Then when , . When , If , then we put in (17), whose left side is less than and equal to , a contradiction. So, .
Finally, holds.

Case 2. The unicyclic graph , where   (), is as follows.
Note that By (2), (12), and (20), we have From Lemmas 5(ii) and 7, So, By a similar discussion to that in Case 1, we have . Thus, .

Lemma 9. Give a sequence of Laplacian spectral radii of even unicyclic graphs as follows: where , , or . Then(i)if ,when , the items of the sequence are equal to each other; that is, when , the sequence is strictly decreasing; when , the sequence is strictly increasing;(ii)if ,when , the items of the sequence are equal to each other; that is, when , the sequence is strictly decreasing; when , the sequence is strictly increasing.

Proof. For any , by (7), we have From (10), and (27) and Lemma 6,
Case 1 (). If , we have by the previous discussion. From (28), the items of the sequence are equal to each other.
If , by Lemma 3, then , where . Note that ; by Lemma 1 and (28), we obtain that the sequence is strictly decreasing.
If , by Lemma 3, then , where . Note that ; by Lemma 1 and (28), we obtain that the sequence is strictly increasing.
Case 2 (). By a similar discussion to that in Case 1, the results of (ii) hold.

Theorem 10. For any , that is, , if is even and the number of is at most , supposing that , then(i)if ,when , are minimal; when , is the smallest; when , is the smallest;(ii)if ,when , are minimal; when , is the smallest; when , is the smallest.

Proof. Given any , for each , let be a rooted tree (with as its root) attached to (), where the order of is . We assume that all trees but, say, , are kept fixed, while (along with its root) can be changed. Suppose that . Let be a vertex belonging to chosen so that and that (the distance between and ) is the largest. By Lemma 4, the Laplacian spectral radius is strictly decreasing when two hanging paths at are replaced by a new hanging path (i.e., the length of the new hanging path is the sum of the lengths of two hanging paths at ). If the same is repeated for other hanging paths to , we get one path attached at (its central vertex is identified with ) whose size is equal to the sum of the lengths of the aforementioned paths. Let be a vertex in , adjacent to and belonging to the (unique) path between and . Note also that . By repeating the same procedure (for any other vertex as ), we arrive at , where the rooted tree becomes a path , so that .
By the same way as with other rooted trees, we arrive at , where every rooted tree (); that is, . By Lemma 4, holds.
If there are at most two trees attached to the cycle , by the previous discussion, we get that has small Laplacian spectral radius among , where two () are nonzero at most. Applying Lemma 8 repeatedly, we know that has the smaller Laplacian spectral radius, where . From Lemma 9, the result holds.

Remark 11. When , that is, , from Theorem 2 and its proof (see [13]), we get the same result as Theorem 10.

Acknowledgments

The author would like to express his sincere gratitude to the referees and the handing editor for the very careful reading of the paper and for the insightful comments and valuable suggestions, which led to improving this paper. The research of the author is supported by the National Natural Science Foundation of China (no. 61303020) and the National Natural Science Foundation of Shanxi Province (no. 2012011019-2).

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