Abstract
Let be the unicyclic graph with vertices obtained by attaching two paths of lengths and at two adjacent vertices of cycle . Let be the unicyclic graph with vertices obtained by attaching paths of lengths at the same vertex of cycle . In this paper, we prove that and are determined by their Laplacian spectra when is even.
1. Introduction
Let be a simple, undirected graph with vertices. Let be the adjacency matrix of and let be the diagonal matrix of vertex degrees of . The matrices and are called the Laplacian matrix and signless Laplacian matrix of , respectively. The multiset of eigenvalues of and are called the A-spectrum and L-spectrum of , respectively. The eigenvalues of and are called the A-eigenvalues and L-eigenvalues of , respectively. We use and to denote the -eigenvalues and the -eigenvalues of , respectively. Two graphs are said to be L-cospectral (A-cospectral) if they have the same -spectrum (-spectrum). A graph is said to be determined by its L-spectrum (A-spectrum) if there is no other nonisomorphic graph L-cospectral (-cospectral) with . Let , , and denote the characteristic polynomials of the adjacency matrix, the Laplacian matrix, and the signless Laplacian matrix of , respectively. As usual, , , and stand for the path, the cycle, and the complete graph with vertices, respectively. Let denote the line graph of . A tree is called starlike if it has exactly one vertex of degree larger than . Let denote the starlike tree with a vertex of degree such that .
For a connected graph with vertices, is called a unicyclic graph if has edges. Which graphs are determined by their spectrum is a difficult problem in the theory of graph spectra. Here, we introduce some results on spectral characterizations of unicyclic graphs. Let be the unicyclic graph with vertices obtained by attaching paths of lengths at the same vertex of cycle (see Figure 1). Haemers et al. [1] proved that is determined by its A-spectrum when is odd, and all are determined by their L-spectra. It is also known that is determined by its A-spectrum when is even [2]. Liu et al. [3] proved that is determined by its L-spectrum. It is known that is determined by its L-spectrum, and is determined by its A-spectrum if is odd (see [4]). Boulet [5] proved that the sun graph is determined by its L-spectrum. Shen and Hou [6] gave a class of unicyclic graphs with even girth that are determined by their L-spectra.
Let be the unicyclic graph with vertices obtained by attaching two paths of lengths and at two adjacent vertices of cycle (see Figure 1). In this paper, we prove that and are determined by their L-spectra when is even.
2. Preliminaries
In this section, we give some lemmas which play important roles throughout this paper.
Lemma 1 (see [7]). Let be a graph. For the adjacency matrix and the Laplacian matrix, the following can be obtained from the spectrum:(i)the number of vertices,(ii)the number of edges.
For the adjacency matrix, the following follows from the spectrum:(iii)the number of closed walks of any length.
For the Laplacian matrix, the following follows from the spectrum:(iv)the number of components,(v)the number of spanning trees.
Lemma 2 (see [8]). For a bipartite graph , one has .
Lemma 3 (see [8]). Let be a graph with vertices and edges. Then
For a graph with vertices, let . Oliveira et al. determined the first four coefficients of as follows.
Lemma 4 (see [9]). Let be a graph with vertices and edges, and let be the degree sequence of . Then where is the number of triangles in .
For a graph , the subdivision graph of , denoted by , is the graph obtained from by inserting a new vertex in each edge of .
Lemma 5 (see [8]). Let be a graph with vertices and edges. Then
Lemma 6 (see [8]). Let be a vertex of , let be the set of all vertices adjacent to , and let be the set of all cycles containing . Then where is the vertex set of .
Lemma 7 (see [10]). Consider .
Lemma 8 (see [1]). Let be a graph with vertices and let be a vertex of . Then .
Lemma 9 (see [5]). Let be a graph with edge set . Then where stands for the degree of vertex .
Lemma 10 (see [11]). For a connected graph with at least two vertices, one has , where denotes the maximum vertex degree of ; equality holds if and only if .
Lemma 11 (see [12]). Let be a connected graph with vertices and let be the second maximum degree of . Then .
Lemma 12 (see [8]). Let be a graph with vertices and let be an edge of . Then .
For a graph , let denote the number of subgraphs of which are isomorphic to graph .
Lemma 13 (see [13]). Let be a graph and let be the number of closed walks of length in . Then
3. Main Results
Lemma 14. Let be a unicyclic graph with vertices, and contains an even cycle . Let be a graph L-cospectral with . Then the following statements hold.(1) is a unicyclic graph with vertices, and the girth of is .(2)The line graphs and are A-cospectral.(3)The subdivision graphs and are A-cospectral, and ().
Proof. By Lemma 1, is a unicyclic graph with vertices, and the girth of is . Since is even, and are bipartite. By Lemma 2, one has . Lemma 3 implies that line graphs and are -cospectral. By Lemma 5, subdivision graphs and are -cospectral, and ().
Theorem 15. The unicyclic graph is determined by its L-spectrum when is even.
Proof. Let be any graph -cospectral with . By Lemma 14, we know that is a unicyclic graph with vertices, the girth of is , and and are -cospectral. By Lemmas 1 and 13, we have . So the maximum degree of does not exceed . Suppose that there are vertices of degree in . From Lemma 4, we have
Solving the above equations, we get . So and have the same degree sequence. Then, one of the following holds.(1) is the unicyclic graph obtained by attaching two paths of lengths and at two nonadjacent vertices of cycle .(2); that is, is the unicyclic graph obtained by attaching two paths of lengths and at two adjacent vertices of cycle .(3) is the graph shown in Figure 2.
Next, we discuss each of these three cases listed above.
Case 1 ( is the unicyclic graph obtained by attaching two paths of lengths and at two nonadjacent vertices of cycle ). Since and are -cospectral, by Lemma 1, and have the same number of closed walks of any length. It is not difficult to see that . By Lemma 13, we have . Note that . If or , then and . If , then and . Hence , a contradiction.
Case 2 ( is the unicyclic graph ). From Lemma 14, we know that the subdivision graphs and (shown in Figure 3) are -cospectral. Let ; from Lemmas 6 and 7, we have
By , we get . By , we get . Hence, or , and are isomorphic.
Case 3 ( is the graph shown in Figure 2). It is well known that the largest -eigenvalue of a path is less than , and the largest -eigenvalue of an even cycle is . Lemma 12 implies that . Let and be the two vertices of degree in (see Figure 2). If and are nonadjacent, there exists an edge of such that . By Lemmas 10 and 2.12, we get , a contradiction to . So and are adjacent.
From Lemma 14, we know that the subdivision graphs and (shown in Figure 4) are -cospectral. Let ; from Lemmas 6 and 7, we have
Since , we have . By , we get , a contradiction to .
Here, we describe a classic method to count the number of closed walks of a given length in a graph (see [2, 13, 14]). For a graph , stands for the number of closed walks of length in and stands for the number of subgraphs of which are isomorphic to graph . Let be the number of closed walks of length of graph which contains all edges of , and denotes the set of all connected subgraphs of such that . Then
Lemma 16. Let and be L-cospectral graphs. If is even, then and are isomorphic.
Proof. If is even, by Lemma 14, and are -cospectral. From Lemma 1, we get for any positive integer . Suppose , . Let . If , by , we know that . For any and , we have . Since , by (10), we get , a contradiction. So we have . Similar to the above arguments, by counting the number of closed walks of length , we can get . Hence and are isomorphic.
Theorem 17. The unicyclic graph is determined by its L-spectrum when is even.
Proof. Let be any graph -cospectral with . By Lemma 14, is a unicyclic graph with vertices, and the girth of is . Let be the vertex of degree in the subdivision graph ; then . Since the largest -eigenvalue of a path is less than , by Lemmas 8 and 14, we get . Suppose is the degree sequence of . By Lemma 11, we have . From Lemmas 9 and 10, we get , , and . By , we have .
If , applying Lemma 4, we have
Since is minimal if and only if for any , the degree sequences of and are both . Lemma 16 implies that and are isomorphic.
If , by and , we get . Suppose that there are three, two, and one in . By Lemma 4, we have
Solving the above equations, we get . From Lemma 4, we have
or is the solution of the above equation. Then or , a contradiction to .
The join of two graphs and , denoted by , is the graph obtained from by joining each vertex of to each vertex of . Some results on spectral characterizations of graphs obtained by join operation can be found in [15–20]. For a unicyclic graph , if is determined by its -spectrum and , then is determined by its -spectrum (cf. [18, Theorem 4.4]). Hence, we can obtain the following two results from Theorems 15 and 17.
Corollary 18. Let . Then is determined by its L-spectrum when is even.
Corollary 19. Let . Then is determined by its L-spectrum when is even.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grant 11371109 and the Fundamental Research Funds for the Central Universities.