JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/268464 268464 Research Article Laplacian Spectral Characterization of Some Unicyclic Graphs Yu Lijun 1 Wang Hui 1 Zhou Jiang 2 Zhang Heping 1 College of Automation, Harbin Engineering University, Harbin 150001 China hrbeu.edu.cn 2 College of Science, Harbin Engineering University, Harbin 150001 China hrbeu.edu.cn 2014 292014 2014 17 06 2014 25 08 2014 3 9 2014 2014 Copyright © 2014 Lijun Yu 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.

Let W ( n ; q , m 1 , m 2 ) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m 1 and m 2 at two adjacent vertices of cycle C q . Let U ( n ; q , m 1 , m 2 , , m s ) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m 1 , m 2 , , m s at the same vertex of cycle C q . In this paper, we prove that W ( n ; q , m 1 , m 2 ) and U ( n ; q , m 1 , m 2 , , m s ) are determined by their Laplacian spectra when q is even.

1. Introduction

Let G be a simple, undirected graph with n vertices. Let A be the adjacency matrix of G and let D be the diagonal matrix of vertex degrees of G . The matrices L = D - A and Q = D + A are called the Laplacian matrix and signless Laplacian matrix of G , respectively. The multiset of eigenvalues of A and L are called the A-spectrum and L-spectrum of G , respectively. The eigenvalues of A and L are called the A-eigenvalues and L-eigenvalues of G , respectively. We use λ 1 ( G ) λ 2 ( G ) λ n ( G ) and μ 1 ( G ) μ 2 ( G ) μ n ( G ) = 0 to denote the A -eigenvalues and the L -eigenvalues of G , respectively. Two graphs are said to be L-cospectral (A-cospectral) if they have the same L -spectrum ( A -spectrum). A graph G is said to be determined by its L-spectrum (A-spectrum) if there is no other nonisomorphic graph L-cospectral ( A -cospectral) with G . Let ϕ A ( G , x ) , ϕ L ( G , x ) , and ϕ Q ( G , x ) denote the characteristic polynomials of the adjacency matrix, the Laplacian matrix, and the signless Laplacian matrix of G , respectively. As usual, P n , C n , and K n stand for the path, the cycle, and the complete graph with n vertices, respectively. Let l ( G ) denote the line graph of G . A tree is called starlike if it has exactly one vertex of degree larger than 2 . Let T a , b , c denote the starlike tree with a vertex v of degree 3 such that T a , b , c - v = P a P b P c .

For a connected graph G with n vertices, G is called a unicyclic graph if G has n 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 U ( n ; q , m 1 , m 2 , , m s ) be the unicyclic graph with n vertices obtained by attaching s paths of lengths m 1 , m 2 , , m s ( m i 1 ) at the same vertex of cycle C q (see Figure 1). Haemers et al.  proved that U ( n ; q , m 1 ) is determined by its A-spectrum when q is odd, and all U ( n ; q , m 1 ) are determined by their L-spectra. It is also known that U ( n ; q , m 1 ) is determined by its A-spectrum when q is even . Liu et al.  proved that U ( n ; q , m 1 , m 2 ) is determined by its L-spectrum. It is known that U ( n ; q , 1,1 , , 1 ) is determined by its L-spectrum, and U ( n ; q , 1,1 , , 1 ) is determined by its A-spectrum if q is odd (see ). Boulet  proved that the sun graph is determined by its L-spectrum. Shen and Hou  gave a class of unicyclic graphs with even girth that are determined by their L-spectra.

Two classes of unicyclic graphs.

Let W ( n ; q , m 1 , m 2 ) be the unicyclic graph with n vertices obtained by attaching two paths of lengths m 1 and m 2 ( m 1 , m 2 1 ) at two adjacent vertices of cycle C q (see Figure 1). In this paper, we prove that W ( n ; q , m 1 , m 2 ) and U ( n ; q , m 1 , m 2 , , m s ) are determined by their L-spectra when q is even.

2. Preliminaries

In this section, we give some lemmas which play important roles throughout this paper.

Lemma 1 (see [<xref ref-type="bibr" rid="B6">7</xref>]).

Let G be a graph. For the adjacency matrix and the Laplacian matrix, the following can be obtained from the spectrum:

the number of vertices,

the number of edges.

For the adjacency matrix, the following follows from the spectrum:

the number of closed walks of any length.

For the Laplacian matrix, the following follows from the spectrum:

the number of components,

the number of spanning trees.

Lemma 2 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

For a bipartite graph G , one has ϕ L ( G , x ) = ϕ Q ( G , x ) .

Lemma 3 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let G be a graph with n vertices and m edges. Then (1) ϕ A ( l ( G ) , x ) = ( x + 2 ) m - n ϕ Q ( G , x + 2 ) .

For a graph G with n vertices, let ϕ L ( G , x ) = l 0 x n + l 1 x n - 1 + + l n . Oliveira et al. determined the first four coefficients of ϕ L ( G , x ) as follows.

Lemma 4 (see [<xref ref-type="bibr" rid="B4">9</xref>]).

Let G be a graph with n vertices and m edges, and let d 1 , d 2 , , d n be the degree sequence of G . Then (2) l 0 = 1 , l 1 = - 2 m = - i = 1 n d i , l 2 = 2 m 2 - m - 1 2 i = 1 n d i 2 , l 3 = 1 3 [ - 4 m 3 + 6 m 2 + 3 m 2 i = 1 n d i 2 - i = 1 n d i 3 - 3 i = 1 n d i 2 + 6 N G ( C 3 ) ] , where N G ( C 3 ) is the number of triangles in G .

For a graph G , the subdivision graph of G , denoted by S ( G ) , is the graph obtained from G by inserting a new vertex in each edge of G .

Lemma 5 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let G be a graph with n vertices and m edges. Then (3) ϕ A ( S ( G ) , x ) = x m - n ϕ Q ( G , x 2 ) .

Lemma 6 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let u be a vertex of G , let N ( u ) be the set of all vertices adjacent to u , and let C ( u ) be the set of all cycles containing u . Then (4) ϕ A ( G , x ) = x ϕ A ( G - u , x ) - v N ( u ) ϕ A ( G - u - v , x ) - 2 Z C ( u ) ϕ A ( G - V ( Z ) , x ) , where V ( Z ) is the vertex set of Z .

Lemma 7 (see [<xref ref-type="bibr" rid="B17">10</xref>]).

Consider ϕ A ( P n , 2 ) = n + 1 .

Lemma 8 (see [<xref ref-type="bibr" rid="B8">1</xref>]).

Let G be a graph with n vertices and let v be a vertex of G . Then λ 1 ( G ) λ 1 ( G - v ) λ 2 ( G ) λ 2 ( G - v ) λ n - 1 ( G - v ) λ n ( G ) .

Lemma 9 (see [<xref ref-type="bibr" rid="B2">5</xref>]).

Let G be a graph with edge set E ( G ) . Then (5) μ 1 ( G ) max { d ( u ) + d ( v ) : u v E ( G ) } , where d ( u ) stands for the degree of vertex u .

Lemma 10 (see [<xref ref-type="bibr" rid="B7">11</xref>]).

For a connected graph G with at least two vertices, one has μ 1 ( G ) Δ ( G ) + 1 , where Δ ( G ) denotes the maximum vertex degree of G ; equality holds if and only if Δ ( G ) = n - 1 .

Lemma 11 (see [<xref ref-type="bibr" rid="B9">12</xref>]).

Let G be a connected graph with n 3 vertices and let d 2 be the second maximum degree of G . Then d 2 μ 2 ( G ) .

Lemma 12 (see [<xref ref-type="bibr" rid="B5">8</xref>]).

Let G be a graph with n vertices and let e be an edge of G . Then μ 1 ( G ) μ 1 ( G - e ) μ 2 ( G ) μ 2 ( G - e ) μ n - 1 ( G - e ) μ n ( G ) = μ n ( G - e ) = 0 .

For a graph G , let N G ( M ) denote the number of subgraphs of G which are isomorphic to graph M .

Lemma 13 (see [<xref ref-type="bibr" rid="B15">13</xref>]).

Let G be a graph and let N G ( k ) be the number of closed walks of length k in G . Then (6) N G ( 3 ) = 6 N G ( C 3 ) , N G ( 5 ) = 30 N G ( C 3 ) + 10 N G ( C 5 ) + 10 N G ( U ( 4 ; 3,1 ) ) .

3. Main Results Lemma 14.

Let G be a unicyclic graph with n vertices, and G contains an even cycle C q . Let H be a graph L-cospectral with G . Then the following statements hold.

H is a unicyclic graph with n vertices, and the girth of H is q .

The line graphs l ( G ) and l ( H ) are A-cospectral.

The subdivision graphs S ( G ) and S ( H ) are A-cospectral, and μ i ( G ) = λ i ( S ( G ) ) ( i = 1,2 , , n ).

Proof.

By Lemma 1, H is a unicyclic graph with n vertices, and the girth of H is q . Since q is even, G and H are bipartite. By Lemma 2, one has ϕ Q ( G , x ) = ϕ L ( G , x ) = ϕ L ( H , x ) = ϕ Q ( H , x ) . Lemma 3 implies that line graphs l ( G ) and l ( H ) are A -cospectral. By Lemma 5, subdivision graphs S ( G ) and S ( H ) are A -cospectral, and μ i ( G ) = λ i ( S ( G ) ) ( i = 1,2 , , n ).

Theorem 15.

The unicyclic graph G = W ( n ; q , m 1 , m 2 ) is determined by its L-spectrum when q is even.

Proof.

Let H be any graph L -cospectral with G . By Lemma 14, we know that H is a unicyclic graph with n vertices, the girth of H is q , and l ( G ) and l ( H ) are A -cospectral. By Lemmas 1 and 13, we have N l ( H ) ( C 3 ) = N l ( G ) ( C 3 ) = 2 . So the maximum degree of H does not exceed 3 . Suppose that there are a i vertices of degree i ( i = 1,2 , 3 ) in H . From Lemma 4, we have (7) i = 1 3 a i = n , i = 1 3 i a i = 2 n , i = 1 3 i 2 a i = 2 × 3 2 + 4 ( n - 4 ) + 2 = 4 n + 4 . Solving the above equations, we get a 1 = 2 , a 2 = n - 4 , a 3 = 2 . So H and G have the same degree sequence. Then, one of the following holds.

H is the unicyclic graph obtained by attaching two paths of lengths l 1 and l 2 at two nonadjacent vertices of cycle C q .

H = W ( n ; q , l 1 , l 2 ) ; that is, H is the unicyclic graph obtained by attaching two paths of lengths l 1 and l 2 at two adjacent vertices of cycle C q .

H is the graph shown in Figure 2.

Next, we discuss each of these three cases listed above.

Case  1 ( H is the unicyclic graph obtained by attaching two paths of lengths l 1 and l 2 at two nonadjacent vertices of cycle C q ). Since l ( G ) and l ( H ) are A -cospectral, by Lemma 1, l ( G ) and l ( H ) have the same number of closed walks of any length. It is not difficult to see that N l ( G ) ( C 5 ) = N l ( H ) ( C 5 ) . By Lemma 13, we have N l ( H ) ( U ( 4 ; 3,1 ) ) = N l ( G ) ( U ( 4 ; 3,1 ) ) . Note that m 1 + m 2 + q = l 1 + l 2 + q = n . If m 1 2 or m 2 2 , then N l ( G ) ( U ( 4 ; 3,1 ) ) 7 and N l ( H ) ( U ( 4 ; 3,1 ) ) 6 . If m 1 = m 2 = 1 , then N l ( G ) ( U ( 4 ; 3,1 ) ) = 6 and N l ( H ) ( U ( 4 ; 3,1 ) ) = 4 . Hence N l ( H ) ( U ( 4 ; 3,1 ) ) N l ( G ) ( U ( 4 ; 3,1 ) ) , a contradiction.

Case  2 ( H is the unicyclic graph W ( n ; q , l 1 , l 2 ) ). From Lemma 14, we know that the subdivision graphs S ( G ) and S ( H ) (shown in Figure 3) are A -cospectral. Let p f = ϕ A ( P f , x ) ; from Lemmas 6 and 7, we have (8) ϕ A ( S ( G ) , x ) = x p 2 m 1 + 2 m 2 + 2 q - 1 - ( p 2 m 1 p 2 q - 2 + 2 m 2 + p 2 m 2 p 2 q - 2 + 2 m 1 ) - 2 p 2 m 1 p 2 m 2 , ϕ A ( S ( G ) , 2 ) = 2 ( 2 m 1 + 2 m 2 + 2 q ) - ( 2 m 1 + 1 ) ( 2 q + 2 m 2 - 1 ) - ( 2 m 2 + 1 ) ( 2 q + 2 m 1 - 1 ) - 2 ( 2 m 1 + 1 ) ( 2 m 2 + 1 ) = - 4 ( m 1 q + m 2 q + 4 m 1 m 2 ) , ϕ A ( S ( H ) , x ) = x p 2 l 1 + 2 l 2 + 2 q - 1 - ( p 2 l 1 p 2 q - 2 + 2 l 2 + p 2 l 2 p 2 q - 2 + 2 l 1 ) - 2 p 2 l 1 p 2 l 2 , ϕ A ( S ( H ) , 2 ) = 2 ( 2 l 1 + 2 l 2 + 2 q ) - ( 2 l 1 + 1 ) ( 2 q + 2 l 2 - 1 ) - ( 2 l 2 + 1 ) ( 2 q + 2 l 1 - 1 ) - 2 ( 2 l 1 + 1 ) ( 2 l 2 + 1 ) = - 4 ( l 1 q + l 2 q + 4 l 1 l 2 ) .

By ϕ A ( S ( G ) , 2 ) = ϕ A ( S ( H ) , 2 ) , we get - 4 ( m 1 q + m 2 q + 4 m 1 m 2 ) = - 4 ( l 1 q + l 2 q + 4 l 1 l 2 ) . By m 1 + m 2 + q = l 1 + l 2 + q = n , we get m 1 m 2 = l 1 l 2 . Hence, m 1 = l 1 , m 2 = l 2 or m 1 = l 2 , m 2 = l 1 , G and H are isomorphic.

Case  3 ( H is the graph shown in Figure 2). It is well known that the largest L -eigenvalue of a path is less than 4 , and the largest L -eigenvalue of an even cycle is 4 . Lemma 12 implies that μ 2 ( G ) < 4 . Let u and v be the two vertices of degree 3 in H (see Figure 2). If u and v are nonadjacent, there exists an edge e of H such that H - e = C q T l 1 , l 2 , n - l 1 - l 2 - q - 1 . By Lemmas 10 and 2.12, we get μ 2 ( H ) 4 , a contradiction to μ 2 ( G ) < 4 . So u and v are adjacent.

From Lemma 14, we know that the subdivision graphs S ( G ) and S ( H ) (shown in Figure 4) are A -cospectral. Let p f = ϕ A ( P f , x ) ; from Lemmas 6 and 7, we have (9) ϕ A ( S ( G ) , x ) = x p 2 m 1 + 2 m 2 + 2 q - 1 - ( p 2 m 1 p 2 q - 2 + 2 m 2 + p 2 m 2 p 2 q - 2 + 2 m 1 ) - 2 p 2 m 1 p 2 m 2 , ϕ A ( S ( G ) , 2 ) = 2 ( 2 m 1 + 2 m 2 + 2 q ) - ( 2 m 1 + 1 ) ( 2 q + 2 m 2 - 1 ) - ( 2 m 2 + 1 ) ( 2 q + 2 m 1 - 1 ) - 2 ( 2 m 1 + 1 ) ( 2 m 2 + 1 ) = - 4 ( m 1 q + m 2 q + 4 m 1 m 2 ) , ϕ A ( S ( H ) , x ) = x p 2 q - 1 ϕ A ( T 1,2 l 1 , 2 l 2 , x ) - ( p 2 q - 1 p 2 l 1 + 2 l 2 + 1 + 2 p 2 q - 2 ϕ A ( T 1,2 l 1 , 2 l 2 , x ) ) - 2 ϕ A ( T 1,2 l 1 , 2 l 2 , x ) , ϕ A ( S ( H ) , 2 ) = 2 × 2 q ϕ A ( T 1,2 l 1 , 2 l 2 , 2 ) - [ 2 q ( 2 l 1 + 2 l 2 + 2 ) + 2 ( 2 q - 1 ) ϕ A ( T 1,2 l 1 , 2 l 2 , 2 ) ] - 2 ϕ A ( T 1,2 l 1 , 2 l 2 , 2 ) = - 4 q ( l 1 + l 2 + 1 ) .

Since ϕ A ( S ( G ) , 2 ) = ϕ A ( S ( H ) , 2 ) , we have - 4 q ( l 1 + l 2 + 1 ) = - 4 ( m 1 q + m 2 q + 4 m 1 m 2 ) . By l 1 + l 2 + 1 = m 1 + m 2 , we get m 1 m 2 = 0 , a contradiction to m 1 , m 2 > 0 .

Graph H .

Two subdivision graphs.

Two subdivision graphs.

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 G , N G ( k ) stands for the number of closed walks of length k in G and N G ( M ) stands for the number of subgraphs of G which are isomorphic to graph M . Let ω k ( M ) be the number of closed walks of length k of graph M which contains all edges of M , and M k ( G ) denotes the set of all connected subgraphs M of G such that ω k ( M ) 0 . Then (10) N G ( k ) = M M k ( G ) N G ( M ) ω k ( M ) .

Lemma 16.

Let G = U ( n ; q , m 1 , m 2 , , m s ) and G = U ( n ; q , l 1 , l 2 , , l s ) be L-cospectral graphs. If q is even, then G and G are isomorphic.

Proof.

If q is even, by Lemma 14, l ( G ) and l ( G ) are A -cospectral. From Lemma 1, we get N l ( G ) ( k ) = N l ( G ) ( k ) for any positive integer k . Suppose m 1 m 2 m s , l 1 l 2 l s . Let r i = min { m i , l i } ( i = 1,2 , , s ) . If m 1 l 1 , by m 1 + m 2 + + m s = l 1 + l 2 + + l s , we know that M 2 r 1 + 3 ( l ( G ) ) = M 2 r 1 + 3 ( l ( G ) ) . For any M M 2 r 1 + 3 ( l ( G ) ) and M U ( 3 + r 1 ; 3 , r 1 ) , we have N l ( G ) ( M ) = N l ( G ) ( M ) . Since N l ( G ) ( U ( 3 + r 1 ; 3 , r 1 ) ) N l ( G ) ( U ( 3 + r 1 ; 3 , r 1 ) ) , by (10), we get N l ( G ) ( 2 r 1 + 3 ) N l ( G ) ( 2 r 1 + 3 ) , a contradiction. So we have m 1 = l 1 . Similar to the above arguments, by counting the number of closed walks of length 2 r i + 3 ( i = 2,3 , , s ) , we can get m i = l i ( i = 2,3 , , s ) . Hence G and G are isomorphic.

Theorem 17.

The unicyclic graph G = U ( n ; q , m 1 , m 2 , , m s ) is determined by its L-spectrum when q is even.

Proof.

Let G be any graph L -cospectral with G . By Lemma 14, G is a unicyclic graph with n vertices, and the girth of G is q . Let v be the vertex of degree s + 2 in the subdivision graph S ( G ) = U ( 2 n ; 2 q , 2 m 1 , 2 m 2 , , 2 m s ) ; then S ( G ) - v = P 2 q - 1 P 2 m 1 P 2 m 2 P 2 m s . Since the largest A -eigenvalue of a path is less than 2 , by Lemmas 8 and 14, we get μ 2 ( G ) = λ 2 ( S ( G ) ) < 2 , μ 2 ( G ) < 4 . Suppose d 1 d 2 d n is the degree sequence of G . By Lemma 11, we have d 2 3 . From Lemmas 9 and 10, we get s + 3 < μ 1 ( G ) s + 4 , d 1 + d 2 μ 1 ( G ) > s + 3 , and d 1 + 1 < μ 1 ( G ) s + 4 . By d 2 3 , we have s < d 1 < s + 3 .

If d 1 = s + 2 , applying Lemma 4, we have (11) i = 2 n d i = 2 + 2 + + 2 n - s - 1 + 1 + 1 + + 1 s , i = 2 n d i 2 = 2 2 + 2 2 + + 2 2 n - s - 1 + 1 2 + 1 2 + + 1 2 s . Since i = 2 n d i 2 is minimal if and only if | d i - d j | 1 for any i , j { 2,3 , , n } , the degree sequences of G and G are both s + 2 , 2,2 , , 2 n - s - 1 , 1,1 , , 1 s . Lemma 16 implies that G and G are isomorphic.

If d 1 = s + 1 , by d 1 + d 2 > s + 3 and d 2 < 4 , we get d 2 = 3 . Suppose that there are a 3 three, a 2 two, and a 1 one in d 2 , d 3 , , d n . By Lemma 4, we have (12) i = 1 3 a i + 1 = n , i = 1 3 i a i + ( s + 1 ) = s + 2 ( n - s - 1 ) + ( s + 2 ) , i = 1 3 i 2 a i + ( s + 1 ) 2 = s + 4 ( n - s - 1 ) + ( s + 2 ) 2 . Solving the above equations, we get a 1 = 2 s - 1 , a 2 = n - 3 s , a 3 = s . From Lemma 4, we have (13) i = 1 3 i 3 a i + ( s + 1 ) 3 = s + 8 ( n - s - 1 ) + ( s + 2 ) 3 . s = 0 or s = 1 is the solution of the above equation. Then d 1 = 1 or d 1 = 2 , a contradiction to d 2 = 3 .

The join of two graphs G and H , denoted by G × H , is the graph obtained from G H by joining each vertex of G to each vertex of H . Some results on spectral characterizations of graphs obtained by join operation can be found in . For a unicyclic graph G , if G is determined by its L -spectrum and G C 6 , then G × K r is determined by its L -spectrum (cf. [18, Theorem 4.4]). Hence, we can obtain the following two results from Theorems 15 and 17.

Corollary 18.

Let G = W ( n ; q , m 1 , m 2 ) . Then G × K r is determined by its L-spectrum when q is even.

Corollary 19.

Let G = U ( n ; q , m 1 , m 2 , , m s ) . Then G × K r is determined by its L-spectrum when q 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.

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