On the spectral radius of block graphs having all their blocks of the same size

块图补图的谱半径与距离谱半径

The spectral radius and the distance spectral radius of complements of block graphs

本文确定了在团树补图和块图补图中，谱半径与距离谱半径达到极值的图。

On the spectral radius of graphs: nonregular distance-hereditary graphs with given edge-connectivity, graphs with tree-width $k$ and block graphs with prescribed independence number $α$

The edge-connectivity of a graph is the minimum number of edges whose deletion disconnects the graph. Let $Δ(G)$ the maximum degree of a graph $G$ and let $ρ(G)$ be the spectral radius of $G$. In this article we present a lower bound for $Δ(G)-ρ(G)$ in terms of the edge connectivity of $G$, where $G$ is a nonregular distance-hereditary graph. We also prove that $ρ(G)$ reaches the maximum at a unique graph in $\mathcal G$, when $\vert V(G)\vert = n$, and $\mathcal G$ either is in the class of graphs with bounded tree-width or is in the class of block graphs with prescribed independence number.

Block matrices and Guo's Index for block circulant matrices with circulant blocks

In this paper we deal with circulant and partitioned into $n$-by-$n$ circulant blocks matrices and introduce spectral results concerning this class of matrices. The problem of finding lists of complex numbers corresponding to a set of eigenvalues of a nonnegative block matrix with circulant blocks is treated. Along the paper we call realizable list if its elements are the eigenvalues of a nonnegative matrix. The Guo's index $λ_0$ of a realizable list is the minimum spectral radius such that the list (up to the initial spectral radius) together with $λ_0$ is realizable. The Guo's index of block…

On the Maximum ABC Spectral Radius of Connected Graphs and Trees

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$ and $m=|E|$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $Δ=\max_{1\leq i \leq n} d_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times n}$, where $m_{ij}=\sqrt{(d_i + d_j -2)/(d_i d_j)}$ if $v_i v_j \in E$, and 0 otherwise. The largest eigenvalue of $M(G)$ is called the ABC spectral radius of $G$, denoted by $ρ_{ABC}(G)$. Recently, this graph invariant has attracted some attentions. We prove that $ρ_{ABC}(G) \leq \sqrt{Δ+(2m-n+1)/Δ-2}$. As an application, the unique tree with $n \geq 4$ vertices …

Spectral isometries onto algebras having a separating family of finite-dimensional irreducible representations

We prove that if $\mathcal{A}$ is a complex, unital semisimple Banach algebra and $\mathcal{B}$ is a complex, unital Banach algebra having a separating family of finite-dimensional irreducible representations, then any unital linear operator from $\mathcal{A}$ onto $\mathcal{B}$ which preserves the spectral radius is a Jordan morphism.

A note on J-positive block operator matrices

We study basic spectral properties of J-self-adjoint $2\times 2$ block operator matrices. Using the linear resolvent growth condition, we obtain simple necessary conditions for the regularity of the critical point $\infty$. In particular, we present simple examples of operators having the singular critical point $\infty$. Also, we apply our results to the linearized operator arising in the study of soliton type solutions to the nonlinear relativistic Ginzburg-Landau equation.