On the spectral radii of unicyclic graphs with fixed matching number

In this paper, the first five sharp upper bounds on the spectral radii of unicyclic graphs with fixed matching number are presented. The first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings are also given, respectively.     

On minimally 2-(edge)-connected graphs with extremal spectral radius

A graph is minimally 2-(edge)-connected if it is 2-(edge)-connected and deleting any arbitrary chosen edge always leaves a graph which is not 2-(edge)-connected. In this paper, we completely characterize the minimally 2-(edge)-connected graphs having the largest and the smallest spectral radius, respectively.     

Halin图谱半径的新上界及极图

利用移接变形的方法再结合特征值的计算技巧刻画出Halin图中谱半径达到第二大的极图,从而得到除轮图以外的Halin图的谱半径的上界以及极图.     

A sharp upper bound on the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain an upper bound on the spectral radius of the adjacency matrix and characterize graphs for which the bound is attained.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.     

The Laplacian spectral radius of graphs on surfaces

Let G be an n-vertex (n?3) simple graph embeddable on a surface of Euler genus γ (the number of crosscaps plus twice the number of handles). Denote by Δ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows:

(i)

     

On an upper bound of the spectral radius of graphs

We give some upper bounds for the spectral radius of bipartite graph and graph, which improve the result in Hong’s Paper [Y. Hong, J.-L. Shu, K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B 81 (2001) 177-183].     

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.     

On the spectral radius of a nonnegative centrosymmetric matrix

In this paper, we discuss the spectral radius of nonnegative centrosymmetric matrices. By using the centrosymmetric structure, we establish some estimations of the spectral radius.     

Extremal graph characterization from the bounds of the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained.     

Average eccentricity,k-packing and k-domination in graphs

Let $G$ be a connected graph. The eccentricity $e\left(v\right)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity $\mathrm{avec}\left(G\right)$ of $G$ is defined as the average of the eccentricities of the vertices of $G$, i.e., as $\frac{1}{\left|V\right|}{\sum }_{v\in V}e\left(v\right)$, where $V$ is the vertex set of $G$. For $k\in \mathbb{N}$, the $k$-packing number of $G$ is the largest cardinality of a set of vertices of $G$ whose pairwise distance is greater than $k$. A $k$-dominating set of $G$ is a set $S$ of vertices such that every vertex of $G$ is within distance $k$ from some vertex of $S$. The $k$-domination number (connected $k$-domination number) of $G$ is the minimum cardinality of a $k$-dominating set (of a $k$-dominating set that induces a connected subgraph) of $G$. For $k=1$, the $k$-packing number, the $k$-domination number and the connected $k$-domination number are the independence number, the domination number and the connected domination number, respectively. In this paper we present upper bounds on the average eccentricity of graphs in terms of order and either $k$-packing number, $k$-domination number or connected $k$-domination number.     

Matrix polynomials with spectral radius equal to the numerical radius

In this paper we study a class of matrix polynomials with the property that spectral radius and numerical radius coincide. Special attention is paid to the spectrum on the boundary of the numerical range.