The Signless Laplacian Spectral Characterization of Strongly Connected Bicyclic Digraphs

DOI：10.3770/j.issn:2095-2651.2016.01.001

 作者 单位 席维鸽 西北工业大学理学院应用数学系, 陕西 西安 710072 王力工 西北工业大学理学院应用数学系, 陕西 西安 710072

设$\overrightarrow{G}$ 是一个强连通双圈有向图, $A(\overrightarrow{G})$是其邻接矩阵.设$D(\overrightarrow{G})$ 是$\overrightarrow{G}$的顶点出度的对角矩阵, $Q(\overrightarrow{G})=D(\overrightarrow{G})+A(\overrightarrow{G})$是$\overrightarrow{G}$ 的无符号拉普拉斯矩阵. $Q(\overrightarrow{G})$的谱半径称为$\overrightarrow{G}$的无符号拉普拉斯谱半径.在这篇文章中, 确定了在所有强连通双圈有向图中达到最大或最小无符号拉普拉斯谱半径的唯一有向图. 此外,还证明了任意一个强连通双圈有向图是由它的无符号拉普拉斯谱所确定的.

Let $\overrightarrow{G}$ be a digraph and $A(\overrightarrow{G})$ be the adjacency matrix of $\overrightarrow{G}$. Let $D(\overrightarrow{G})$ be the diagonal matrix with outdegrees of vertices of $\overrightarrow{G}$ and $Q(\overrightarrow{G})=D(\overrightarrow{G})+A(\overrightarrow{G})$ be the signless Laplacian matrix of $\overrightarrow{G}$. The spectral radius of $Q(\overrightarrow{G})$ is called the signless Laplacian spectral radius of $\overrightarrow{G}$. In this paper, we determine the unique digraph which attains the maximum (or minimum) signless Laplacian spectral radius among all strongly connected bicyclic digraphs. Furthermore, we prove that any strongly connected bicyclic digraph is determined by the signless Laplacian spectrum.