Laplacian Spectral Characterization of Some Unicyclic Graphs

Let G be a simple, undirected graph with n vertices. Let A be the adjacency matrix ofG and letD 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 Leigenvalues of G, respectively. We use λ 1 (G) ⩾ λ 2 (G) ⩾ ⋅ ⋅ ⋅ ⩾


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 Leigenvalues of , respectively.We use  1 () ⩾  2 () ⩾ ⋅ ⋅ ⋅ ⩾   () and  1 () ⩾  2 () ⩾ ⋅ ⋅ ⋅ ⩾   () = 0 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 ℓ() denote the line graph of .A tree is called starlike if it has exactly one vertex of degree larger than 2. Let  ,, denote the starlike tree with a vertex V of degree 3 such that  ,, − V =   ∪   ∪   .
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 (; ,  1 ,  2 , . . .,   ) be the unicyclic graph with  vertices obtained by attaching  paths of lengths  1 ,  2 , . . .,   (  ⩾ 1) at the same vertex of cycle   (see Figure 1).Haemers et al. [1] proved that (; ,  1 ) is determined by its A-spectrum when  is odd, and all (; ,  1 ) are determined by their L-spectra.It is also known that (; ,  1 ) is determined by its Aspectrum when  is even [2].Liu et al. [3] proved that (; ,  1 ,  2 ) is determined by its L-spectrum.It is known that (; , 1, 1, . . ., 1) is determined by its L-spectrum, and (; , 1, 1, . . ., 1) 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.

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 4 (see [9]).Let  be a graph with  vertices and  edges, and let  1 ,  2 , . . .,   be the degree sequence of .Then where   ( 3 ) 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 8 (see [1]).Let  be a graph with  vertices and let V be a vertex of .
Lemma 9 (see [5]).Let  be a graph with edge set ().Then where () stands for the degree of vertex .
Lemma 12 (see [8]).Let  be a graph with  vertices and let  be an edge of .
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

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 ℓ() and ℓ() are A-cospectral. ( So the maximum degree of  does not exceed 3. Suppose that there are   vertices of degree  ( = 1, 2, 3) 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  1 and  2 at two nonadjacent vertices of cycle   .
(2)  = (; ,  1 ,  2 ); that is,  is the unicyclic graph obtained by attaching two paths of lengths  1 and  2 at two adjacent vertices of cycle   .
(3)  is the graph shown in Figure 2.
Next, we discuss each of these three cases listed above.
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   ((), 2) =   ((), 2), we have 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   () ̸ = 0. Then   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][16][17][18][19][20].For a unicyclic graph , if  is determined by its -spectrum and  ̸ =  6 , then  ×   is determined by its -spectrum (cf.[18,Theorem 4.4]).Hence, we can obtain the following two results from Theorems 15 and 17.