The Laplacian Spread of Bicyclic Graphs  

作　　者：Yi Zheng FAN Shuang Dong LI Ying Ying TAN 

机构地区：[1]School of Mathematical Sciences, Anhui University, Anhui 230039, P. R. China [2]Department of Mathematics & Physics, Anhui University of Architecture, Anhui 230022, P. R. China

出　　处：《Journal of Mathematical Research and Exposition》2010年第1期17-28,共12页数学研究与评论（英文版）

基　　金：Supported by the National Natural Science Foundation of China (Grant No.10601001);the Natural Science Foundation of Anhui Province (Grant No.070412065);Project of Anhui Province for Young Teachers Research Supportin Universities (Grant No.2008jql083);Natural Science Foundation of Department of Education of Anhui Province(Grant No.2005kj005zd);Project of Anhui University on Leading Researchers Construction;Foundation of Innovation Team on Basic Mathematics of Anhui University

摘　　要：The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.

关 键 词：bicyclic graph Laplacian matrix spread. 

分 类 号：O151.21[理学—数学] TP391.41[理学—基础数学]