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.     

Spectra of Corona Based on the Total Graph

For two simple connected graphs $G_1$ and $G_2$, we introduce a new graph operation called the total corona $G_1⊛G_2$ on $G_1$ and $G_2$ involving the total graph of $G_1.$ Subsequently, the adjacency (respectively, Laplacian and signless Laplacian) spectra of $G_1⊛G_2$ are determined in terms of these of a regular graph $G_1$ and an arbitrary graph $G_2.$ As applications, we construct infinitely many pairs of adjacency (respectively, Laplacian and signless Laplacian) cospectral graphs. Besides we also compute the number of spanning trees of $G_1⊛G_2.$     

Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity

In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity.     

Q整图新类

对于一个简单图G, 方阵Q(G)=D(G)+A(G)称为G的无符号拉普拉斯矩阵,其中D(G)和A(G)分别为G的度对角矩阵和邻接矩阵. 一个图是Q整图是指该图的无符号拉普拉斯矩阵的特征值全部为整数.首先通过Stanic 得到的六个顶点数目较小的Q整图,构造出了六类具有无穷多个的非正则的Q整图. 进而,通过图的笛卡尔积运算得到了很多的Q整图类. 最后, 得到了一些正则的Q整图.     

Some results on the signless Laplacians of graphs

In this paper we give two results concerning the signless Laplacian spectra of simple graphs. Firstly, we give a combinatorial expression for the fourth coefficient of the (signless Laplacian) characteristic polynomial of a graph. Secondly, we consider limit points for the (signless Laplacian) eigenvalues and we prove that each non-negative real number is a limit point for (signless Laplacian) eigenvalue of graphs.     

On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

图的无号Laplace矩阵的最大特征值

The signless Laplacian matrix of a graph is the sum of its diagonal matrix of vertex degrees and its adjacency matrix. Li and Feng gave some basic results on the largest eigenvalue and characteristic polynomial of adjacency matrix of a graph in 1979. In this paper, we translate these results into the signless Laplacian matrix of a graph and obtain the similar results.     

Bounds for the signless Laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.     

完全多部图的无符号Laplacian特征多项式(英文)

For a simple graph G,let matrix Q（G）=D（G） + A（G） be it’s signless Laplacian matrix and Q G （λ）=det（λI Q） it’s signless Laplacian characteristic polynomial,where D（G） denotes the diagonal matrix of vertex degrees of G,A（G） denotes its adjacency matrix of G.If all eigenvalues of Q G （λ） are integral,then the graph G is called Q-integral.In this paper,we obtain that the signless Laplacian characteristic polynomials of the complete multi-partite graphs G=K（n₁,n₂,···,n_t）.We prove that the complete t-partite graphs K（n,n,···,n）t are Q-integral and give a necessary and sufficient condition for the complete multipartite graphs K（m,···,m）s（n,···,n）t to be Q-integral.We also obtain that the signless Laplacian characteristic polynomials of the complete multipartite graphs K（m,···,m,）s1（n,···,n,）s2（l,···,l）s3.     

On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively.     

Sharp bounds on the spectral radius of a nonnegative matrix

We give upper and lower bounds for the spectral radius of a nonnegative matrix using its row sums and characterize the equality cases if the matrix is irreducible. Then we apply these bounds to various matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix. Some known results in the literature are generalized and improved.     

哈密顿图的无符号拉普拉斯谱半径条件

令A(G)=(a_(ij))_(n×n)是简单图G的邻接矩阵,其中若v_i-v_j,则a_(ij)=1,否则a_(ij)=0.设D(G)是度对角矩阵,其(i,i)位置是图G的顶点v_i的度.矩阵Q(G)=D(G)+A(G)表示无符号拉普拉斯矩阵.Q(G)的最大特征根称作图G的无符号拉普拉斯谱半径,用q(G)表示.Liu,Shiu and Xue[R.Liu,W.Shui,J.Xue,Sufficient spectral conditions on Hamiltonian and traceable graphs,Linear Algebra Appl.467(2015)254-255]指出:可以通过复杂的结构分析和排除更多的例外图,当q(G)≥2n-6+4/(n-1)时,则G是哈密顿的.作为论断的有力补充,给出了图是哈密顿图的一个稍弱的充分谱条件,并给出了详细的证明和例外图.     

Spectra of graphs obtained by a generalization of the join graph operation

Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.     

双圈图的距离无符号拉普拉斯谱半径

连通图$G$的距离无符号拉普拉斯矩阵定义为$\mathcal{Q}(G)=Tr(G)+D(G)$, 其中$Tr(G)$和$D(G)$分别为连通图$G$的点传输矩阵和距离矩阵. 图$G$的距离无符号拉普拉斯矩阵的最大特征值称为$G$的距离无符号拉普拉斯谱半径. 本文确定了给定点数的双圈图中具有最大的距离无符号拉普拉斯谱半径的图.     

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.     

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.     

双圈图的无符号拉普拉斯谱半径和谱展

The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.     

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.     

A sharp upper bound for the spectral radius of a nonnegative matrix and applications

We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.     

可迹图的谱充分条件

设G是一个简单图,A(G),Q(G)以及Q(G)分别为G的邻接矩阵,无符号拉普拉斯矩阵以及距离无符号拉普拉斯矩阵,其最大特征值分别称为G的谱半径,无符号拉普拉斯谱半径以及距离无符号拉普拉斯谱半径.如果图G中有一条包含G中所有顶点的路,则称这条路为哈密顿路;如果图G含有哈密顿路,则称G为可迹图;如果图G含有从任意一点出发的哈密顿路,则称G从任意一点出发都是可迹的.主要研究利用图G的谱半径,无符号拉普拉斯谱半径,以及距离无符号拉普拉斯谱半径,分别给出图G从任意一点出发都是可迹的充分条件.