浅析有向图的特征值

摘 要

本论文首先阐述了有向图、连通图、矩阵表示形式（邻接矩阵）、矩阵特征值、矩阵的谱和偶图等基本概念。接着重点叙述了有向图的特征值的主要结论及其证明，即是Perron-Frobenius定理、Levy-Desplanques定理、Gerschgorin圆盘定理、Brauer定理和Brualdi定理等，便于对有向图的特征值的理解和掌握。最后描述了有向图及其特征值在竞技比赛中的应用，即是通过在单循环比赛中排列名次的实例，表明有向图的特征值在实际应用中的重要性。在研究有向图的特征值的过程中，都要把有向图化为矩阵的形式，再研究矩阵的特征值。对于高阶矩阵，很难直接求出它们的特征值，于是，对有向图的特征值的估值是1个重要的课题，本文对此进行了研究。

关键字:有向图；矩阵；特征值；圆盘；竞赛图。

Abstract

This article first expatiate on basic concepts that digraph、the connect graph、matrix denotation form(adjacency matrix)、eigenvalue of matrix、spectrum of matrix and the pear graph and so on. Follow emphase to depiction on mostly conclusion and prove that eigenvalues of digraphs,namely be Perron-Frobenius theorem、Levy-Desplanques theorem、Gerschgorin disc theorems、Brauer theorem and Brualdi theorem and so on,easy to understand and predominate with eigenvalues of digraphs.

Finally,describe on eigenvalues and digraphs applications in the athletics match, namely be pass example arrange place in a competition in the single circle match,indicate eigenvalues and digraphs essentiality in the practice applications. At the research course with eigenvalues of digraphs,all need to hold digraph melt into form of matrix,research eigenvalue of matrix again.For high rank matrix,very hard directness get hold of their eigenvalue,and then,it is one important task that appraise cost with eigenvalues of digraphs,this text withal put up research.

Key words:Digraph; Matrix; Eigenvalue; Disc; Tournament.