On Edge Singularity and Eigenvectors of Mixed Graphs  

作　　者：Ying Ying TAN Yi Zheng FAN 

机构地区：[1]School of Mathematics and Computation Sciences, Anhui University,Hefei 230039, P. R. China [2]Department of Mathematics and Physics, Anhui Institute of Architecture and Industry, Hefei 230022, Po R. China

出　　处：《Acta Mathematica Sinica,English Series》2008年第1期139-146,共8页数学学报（英文版）

基　　金：National Natural Science Foundation of China (10601001);Anhui Provincial Natural Science Foundation (050460102);NSF of Department of Education of Anhui province (2004kj027,2005kj005zd);Foundation of Anhui Institute of Architecture and Industry (200510307);Foundation of Innovation Team on Basic Mathematics of Anhui University;Foundation of Talents Group Construction of Anhui University

摘　　要：Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.Let G be a mixed glaph which is obtained from an undirected graph by orienting some of its edges. The eigenvalues and eigenvectors of G are, respectively, defined to be those of the Laplacian matrix L（G） of G. As L（G） is positive semidefinite, the singularity of L（G） is determined by its least eigenvalue λ1 （G）. This paper introduces a new parameter edge singularity εs（G） that reflects the singularity of L（G）, which is the minimum number of edges of G whose deletion yields that all the components of the resulting graph are singular. We give some inequalities between εs（G） and λ1 （G） （and other parameters） of G. In the case of εs（G） = 1, we obtain a property on the structure of the eigenvectors of G corresponding to λ1 （G）, which is similar to the property of Fiedler vectors of a simple graph given by Fiedler.

关 键 词：mixed graphs edge singularity Laplacian eigenvectors 

分 类 号：O14[理学—数学]