有向图的拉普拉斯谱半径的几个上界

Several Upper Bounds for the Laplacian Spectral Radius of Digraphs

设G=(V(G),E(G))是一个简单有向图具有顶点集V(G)={v_1,v_2,…,v_n)和弧集E(G).用d_i^+表示顶点v_i的出度.设A(G)是有向图G的邻接矩阵和D(G)=diag(d_1^+,d_2^+,…,d_n^+)有向图G的顶点出度对角矩阵,则称L(G)=D(G)-A(G)为有向图G的拉普拉斯矩阵.L(G)的谱半径称作有向图G的拉普拉斯谱半径,用A(G)表示.在这篇文章中,给出了关于A(G)的一些上界,进而一些关于λ(G)涉及有向图G的出度和二次平均出度的上界也被得到.最后,我们举例对这些上界进行了比较.

Let G = （V（G）,E（G）） be a simple digraph with vertex set V（G） = {v1,v2…vn} and arc set E（G）. Denote the outdegree of the vertex vi by d＋. Let A（G） be the adjacency matrix of G and D（G） = diag（d1＋,d2＋,…,dn＋） be the diagonal matrix with outdegrees of the vertices of G. Then we call L（G） = D（G） A（G） the Laplacian matrix of G. The spectral radius of L（G） is called the Laplacian spectral radius of G, denoted by λ（G）. In this paper, we give some upper bounds on λ（G）. Furthermore, some upper bounds on λ（G） involving outdegrees and average 2-outdegrees of the vertices of G are also obtained. Finally, we give an example to compare the bounds.