Several sharp upper bounds for the largest laplacian eigenvalue of a graph

作     者：Tian-fei WANG 

作者机构：Department of Mathematics Leshan Teachers College Leshan 614004 China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2007年第50卷第12期

页      面：1755-1764页

核心收录：

学科分类：07[理学] 08[工学] 0701[理学-数学] 

基　　金：This work was supported by the Natural Science Foundation of Sichuan Province (Grant No.2006C040) 

主　　题：Laplacian matrix, the largest eigenvalue, similar matrix 

摘      要：Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively. In this paper, we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D-1/2 KD 1/2 and U-1/2 BU 1/2, respectively, where D is the degree diagonal matrix of G and U=diag(dudv: uv ∈ E(G)).And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G. Moreover, we determine all extremal graphs which achieve these upper bounds. Finally,some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.