含有割点的图的最小特征值

Let ψ be a certain set of graphs.A graph is called a minimizing graph in the set ψ if its least eigenvalue attains the minimum among all graphs in ψ.In this paper,we determine the unique minimizing graph in ψn,where ψn denotes the set of connected graphs of order n with cut vertices.     

On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

On Edge Singularity and Eigenvectors of Mixed Graphs

Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.     

Upper bound of the third edge-connectivity of graphs

Let G be a simple connected graph of order n≥6. The third edge-connectivity of G is defined as the minimum cardinality over all the sets of edges, if any, whose deletion disconnects G and every component of the resulting graph has at least 3 vertices. In this paper, we first characterize those graphs whose third-edge connectivity is well defined, then establish the tight upper bound for the third edge-connectivity.     

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.     

Competition Numbers and Phytogeny Numbers of Connected Graphs and Hypergraphs

Let D be a digraph.The competition graph of D is the graph having the same vertex set with D and having an edge joining two different vertices if and only if they have at least one common out-neighbor in D.The phylogeny graph of D is the competition graph of the digraph constructed from D by adding loops at all vertices.The competition/phylogeny number of a graph is the least number of vertices to be added to make the graph a competition/phylogeny graph of an acyclic digraph.In this paper,we show that for any integer k there is a connected graph such that its phylogeny number minus its competition number is greater than k.We get similar results for hypergraphs.     

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.     

BOUNDS OF EIGENVALUES OF A GRAPH

Let G be a simple graph with n vertices.We denote by λ_i(G) the i-th largest eigenvalue of G.In this paper,several results are presented concerning bounds on the eigenvalues of G.In particular,it is shown that -1≤λ_2(G)≤(n-2)/2,and the left hand equality holds if and only if G is a complete graph with at least two vertices;the right hand equality holds if and only if n is even and G?2K_(n/2).     

Graphs isomorphic to their maximum matching graphs

The maximum matching graph M（G） of a graph G is a simple graph whose vertices are the maximum matchings of G and where two maximum matchings are adjacent in M（G） if they differ by exactly one edge. In this paper, we prove that if a graph is isomorphic to its maximum matching graph, then every block of the graph is an odd cycle.     

补图为2-点或2-边连通的图的最小特征值

图的最小特征值定义为图的邻接矩阵的最小特征值,是刻画图结构性质的一个重要代数参数. 在所有给定阶数的补图为2-点或2-边连通的图中, 刻画了最小特征值达到极小的唯一图, 并给出了这类图最小特征值的下界.     

补图具有悬挂点且连通的图的最小特征值

图的最小特征值定义为图的邻接矩阵的最小特征值,它是刻画图的结构性质的重要参数.在给定阶数且补图为具有悬挂点的连通图的图类中,刻画了最小特征值达极小的唯一图,并给出了这类图最小特征值的下界.     

The vertex (edge) independence number, vertex (edge) cover number and the least eigenvalue of a graph

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all graphs of a fixed order and a given vertex (edge) independence number or vertex (edge) cover number, and get some bounds for the vertex (edge) independence number, vertex (edge) cover number of a graph in terms of the least eigenvalue of the graph.     

The least eigenvalue of a graph with cut vertices

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut vertices, and then obtain a lower bound for the least eigenvalue of a connected graph in terms of the number of cut vertices. During the discussion we also get some results for the spectral radius of a connected bipartite graph and its upper bound.     

The least eigenvalue of a graph with a given domination number

In this paper, we characterize the unique graph whose least eigenvalue achieves the minimum among all graphs with n vertices and domination number γ. Thus we can obtain a lower bound on the least eigenvalue of a graph in terms of the domination number.     

匹配数为2的图的最小特征值

在所有给定阶数且匹配数为2的连通图中,我们刻画了最小特征值达到极小的图,给出了这类图最小特征值的下界.     

The Least Eigenvalue of Graphs with Cut Edges

In this paper we characterize the unique graph whose least eigenvalue attains the minimum among all connected graphs of fixed order and given number of cut edges.     

On least eigenvalues and least eigenvectors of real symmetric matrices and graphs

We give an upper bound for the least eigenvalue of a principal submatrix of a real symmetric matrix with zero diagonal, from which we establish an upper bound for the least eigenvalue of a graph when some vertices are removed using the components of the least eigenvector(s). We give lower and upper bounds for the least eigenvalue of a graph when some edges are removed. We also establish bounds for the components of the least eigenvector(s) of a real symmetric matrix and a graph.     

Bicyclic graphs for which the least eigenvalue is minimum

The spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. In this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n. Also, we determine the unique graph with maximum spread in this class for each n?28.     

On the least eigenvalue of a unicyclic mixed graph

Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.