单圈混合图的极大谱半径

摘要

设U*为一个未定向的n个顶点上的单圈混合图,它是由一个三角形在其某个顶点上附加n-3个悬挂边而获得.在文[Largest eigenvalue of a unicyclic mixed graph, Applied Mathematics A Journal of Chinese Universities (Ser. B), 2004, 19(2): 140-148]中,作者证明了:在相差符号同构意下,在所有n个顶点上的单圈混合图中,U*是唯一的达到最大Laplace谱半径的混合图.本文应用非负矩阵的Perron向量,给出上述结论的一个简单的证明.%Let U* be an unoriented unicyclic mixed graph on n vertices which is obtained from a triangle by appending n-3 pendent edges to one of its vertices. In the paper [Largest eigenvalue of a unicyclic mixed graph, Applied Mathematics A Journal of Chinese Universities (Ser. B), 2004, 19(2): 140-148], the authors prove that up to signature isomorphisms U* is the unique graph which maximizes Laplacian spectral radius over all unicyclic mixed graphs on n vertices. In this paper, we use a simple method to prove above result by the Perron vectors of nonnegative matrices.