Laplacian spectral characterization of clover graphs

A clover graph is obtained from a 3-rose graph by attaching a path to the vertex of degree six, where a 3-rose graph consists of three cycles with precisely one common vertex. In this paper, it is proved that all clover graphs are determined by their Laplacian spectra.     

Signless Laplacian spectral characterization of 4-rose graphs

A rose graph with p petals (or p-rose graph) is a graph obtained by taking p cycles with just a vertex in common. In this paper, we prove that all 4-rose graphs are determined by their signless Laplacian spectra.     

On the Laplacian integral tricyclic graphs

A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

On the Laplacian spectral characterization of Π-shape trees

A Π-shape tree is a tree with exactly two vertices having the maximum degree three. In this paper, we classify the Π-shape trees into two types, and complete the spectral characterization for one type. Exactly, we prove that all graphs of this type are determined by their Laplacian spectra with some exceptions. Moreover, we give some L-cospectral mates of some graphs for another type.     

On a signless Laplacian spectral characterization of -shape trees

Let M be an associated matrix of a graph G (the adjacency, Laplacian and signless Laplacian matrix). Two graphs are said to be cospectral with respect to M if they have the same M spectrum. A graph is said to be determined by M spectrum if there is no other non-isomorphic graph with the same spectrum with respect to M. It is shown that T-shape trees are determined by their Laplacian spectra. Moreover among them those are determined by their adjacency spectra are characterized. In this paper, we identify graphs which are cospectral to a given T-shape tree with respect to the signless Laplacian matrix. Subsequently, T-shape trees which are determined by their signless Laplacian spectra are identified.     

The spectral characterization of graphs of index less than 2 with no path as a component

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less than 2. First, we give various infinite families of cospectral graphs with respect to the adjacency matrix. Subsequently, the results will be used to characterize all DS graphs (with respect to the adjacency matrix) of index less than 2 with no path as a component. Moreover, we show that most of these graphs are DS with respect to the Laplacian matrix.     

含偶圈图的Laplacian 谱刻画

设 H(K_{1,5},P_n,C_l)是由路 P_n的两个悬挂点分别粘上星图K_{1,5}的悬挂点和圈 C_l的点所得的单圈图. 若两个二部图是关于Laplacian 矩阵同谱的, 则它们的线图是邻接同谱的, 两个邻接同谱图含有相同数目的同长闭回路. 如果任何一个与图G关于Laplacian 同谱图都与图G 同构, 那么称图G可由其Laplacian 谱确定. 利用图与线图之间的关系证明了H(K_{1,5},P_n,C_4)、H(K_{1,5},P_n,C_6) 由它们的Laplacian谱确定.     

几类多圈图的拉普拉斯谱刻画

设图G是一个简单连通图. 如果任何一个与图G同拉普拉斯谱的图都与图G同构,则称图G是由其拉普拉斯谱确定的. 定义了双圈图\theta_{n}(p_1,p_2,\cdots,p_t) 和m 圈图H_n(m\cdot C_3;p_1,p_2,\cdots,p_t). 证明了双圈图\theta_{n}(p)和\theta_{n}(p,q),三圈图H_n(3\cdot C_3;p)和H_n(3\cdot C_3;p,q)分别是由它们的拉普拉斯谱确定的.     

On the asymptotic behavior of graphs determined by their generalized spectra

In Wang and Xu (2006) [15] and [16] the authors introduced a family of graphs H_n and gave some methods for finding graphs among this family that are determined by their generalized spectra. This paper is a continuation of our previous work. We further show that almost all graphs in H_n are determined by their generalized spectra. This gives some evidences for the conjecture that almost all graphs are determined by their generalized spectra.     

Theorems on partitioned matrices revisited and their applications to graph spectra

Some old results about spectra of partitioned matrices due to Goddard and Schneider or Haynsworth are re-proved. A new result is given for the spectrum of a block-stochastic matrix with the property that each off-diagonal block has equal entries and each diagonal block has equal diagonal entries and equal off-diagonal entries. The result is applied to the study of the spectra of the usual graph matrices by partitioning the vertex set of the graph according to the neighborhood equivalence relation. The concept of a reduced graph matrix is introduced. The question of when n-2 is the second largest signless Laplacian eigenvalue of a connected graph of order n is treated. A recent conjecture posed by Tam, Fan and Zhou on graphs that maximize the signless Laplacian spectral radius over all (not necessarily connected) graphs with given numbers of vertices and edges is refuted. The Laplacian spectrum of a (degree) maximal graph is reconsidered.     

Some results on Laplacian spectral radius of graphs with cut vertices

In this paper, we give some results on Laplacian spectral radius of graphs with cut vertices, and as their applications, we also determine the unique graph with the largest Laplacian spectral radius among all unicyclic graphs with n vertices and diameter d, 3?d?n−3.     

新单圈图H(p,tK_(1,m))的拉普拉斯谱刻画

设图H(p,tK_(1,m))是一个顶点数为p+mt的连通单圈图,它是由圈C_p的依次相邻的t(1≤t≤p)个顶点的每一个顶点分别与星K_(1,m)的中心重合而得到的单圈图.现证明单圈图H(p,pK_(1,5)),H(p,(p-1)K_(1,4))是由它们的拉普拉斯谱确定的,并证明了当p为偶数时,单圈图H(p,2K_(1,4)),H(p,(p-2)K_(1,4)),H(p,(p-3)K_(1,4))也是由它们的拉普拉斯谱确定的.     

Spectral characterization of graphs whose second largest eigenvalue is less than 1

Graphs with second largest eigenvalue λ₂?1 are extensively studied, however, whether they are determined by their adjacency spectra or not is less considered. In this paper we completely characterize all the connected bipartite graphs with λ₂<1 that are determined by their adjacency spectra. In addition, we prove that all the connected non-bipartite graphs with girth no less than 4 and λ₂<1 are determined by their adjacency spectra.     

The signless Laplacian spectral radius of graphs with given degree sequences

In this paper, we investigate the properties of the largest signless Laplacian spectral radius in the set of all simple connected graphs with a given degree sequence. These results are used to characterize the unicyclic graphs that have the largest signless Laplacian spectral radius for a given unicyclic graphic degree sequence. Moreover, all extremal unicyclic graphs having the largest signless Laplacian spectral radius are obtained in the sets of all unicyclic graphs of order n with a specified number of leaves or maximum degree or independence number or matching number.     

On the extremal values of the second largest Q-eigenvalue

We study extremal graphs for the extremal values of the second largest Q-eigenvalue of a connected graph. We first characterize all simple connected graphs with second largest signless Laplacian eigenvalue at most 3. The second part of the present paper is devoted to the study of the graphs that maximize the second largest Q-eigenvalue. We construct families of such graphs and prove that some of theses families are minimal for the fact that they maximize the second largest signless Laplacian eigenvalue.     

一类单圈图的拉普拉斯谱刻画

Let H（n; q, n1, n2, n3, n4） be a unicyclic graph with n vertices containing a cycle Cq and four hanging paths Ph1＋1, Pn2＋1, Pn3＋1 and Pn4＋1 attached at the same vertex of the cycle. In this paper, it is proved that all unicyclic graphs H （n; q, n1, n2, n3, n4） are determined by their Laplacian spectra.     

Non-bipartite Graphs with Third Largest Laplacian Eigenvalue Less Than Three

All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non-bipartite graphs with third largest Laplacian eigenvalue less than three are determined.     

Unoriented Laplacian maximizing graphs are degree maximal

A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u,v of G, the sets N(u)?{v},N(v)?{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.