关于图的最小Q-特征值

研究了基于n阶二部图和s阶完全图构造的一个图类,得到了该图类的无符号拉普拉斯最小特征值(即最小Q-特征值)的一个可达上界为s.基于此,对于任意给定的正整数s和正偶数n,构造了最小Q-特征值为s的一类n+s阶图.另外,对于任意给定的最小度δ和阶数n,在满足2≤δ≤n-1/2条件下,构造了最小Q-特征值为δ-1的一类n阶图.     

Graphs with at most one signless Laplacian eigenvalue exceeding three

We determine all connected graphs with at most one signless Laplacian eigenvalue exceeding three.     

Distance signless Laplacian eigenvalues of graphs

Suppose that the vertex set of a graph G is V(G)=\{v\mathrm{1},v\mathrm{2},\mathrm{...},vn\}. The transmission Tr(vi) (or D_i) of vertex vi is defined to be the sum of distances from vi to all other vertices. Let Tr(G) be the n\times n diagonal matrix with its (i, i)-entry equal to {{\displaystyle Tr}}_G(vi). The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as L(G)=Tr(G)+D(G), where D(G) is the distance matrix of G. In this paper, we give a lower bound on the distance signless Laplacian spectral radius of graphs and characterize graphs for which these bounds are best possible. We obtain a lower bound on the second largest distance signless Laplacian eigenvalue of graphs. Moreover, we present lower bounds on the spread of distance signless Laplacian matrix of graphs and trees, and characterize extremal graphs.     

Some notes on the spectral perturbations of the signless Laplacian of a graph

Let G be a simple graph and let Q（G） be the signless Laplacian matrix of G. In this paper we obtain some results on the spectral perturbation of the matrix Q（G） under an edge addition or an edge contraction.     

On thek-th largest eigenvalue of the Laplacian matrix of a graph

In this paper, we give the upper bound and lower bound ofk-th largest eigenvalue λk of the Laplacian matrix of a graphG in terms of the edge number ofG and the number of spanning trees ofG. This research is supported by the National Natural Science Foundation of China (Grant No.19971086) and the Doctoral Program Foundation of State Education Department of China.     

Laplacian and signless Laplacian Z-eigenvalues of uniform hypergraphs

We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d.     

单圈图和双圈图的最大无符号拉普拉斯分离度

设G是一个n阶简单图,q_{1}(G)\geq q_{2}(G)\geq \cdots \geq q_{n}(G)是其无符号拉普拉斯特征值. 图G的无符号拉普拉斯分离度定义为S_{Q}(G)=q_{1}(G)-q_{2}(G). 确定了n阶单圈图和双圈图的最大的无符号拉普拉斯分离度,并分别刻画了相应的极图.     

Bounds on the index of the signless Laplacian of a graph

Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees Δ=d₁≥d₂≥?≥d_n=δ. The average degree of the neighbor of vertex v_i is . Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)−A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let μ₁(G) denote the index of L(G) and q₁(G) the index of Q(G). We survey upper bounds on μ₁(G) and q₁(G) given in terms of the d_i and m_i, as well as the numbers of common neighbors of pairs of vertices. It is well known that μ₁(G)≤q₁(G). We show that many but not all upper bounds on μ₁(G) are still valid for q₁(G).     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.     

An eigenvalue bound for the Laplacian of a graph

We present a lower bound for the smallest non-zero eigenvalue of the Laplacian of an undirected graph. The bound is primarily useful for graphs with small diameter.     

不含P_t的非二部连通图的最小Q-特征值

对于一个连通图而言,它的最小Q-特征值为零当且仅当它是二部图.图的最小Q-特征值常被用来衡量一个图的非二部程度,因而受到研究者的广泛关注.文中研究了图中存在长路的最小Q-特征值条件,分别确定了最小Q-特征值最小的不含路Pt的非二部单圈图和非二部连通图.     

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.     

The signless Laplacian spectra of the corona and edge corona of two graphs

We give complete information about the signless Laplacian spectrum of the corona of a graph G ₁ and a regular graph G ₂, and complete information about the signless Laplacian spectrum of the edge corona of a connected regular graph G ₁ and a regular graph G ₂.