淺析有向圖的特征值

摘　要

本論文首先闡述了有向圖、連通圖、矩陣表示形式（鄰接矩陣）、矩陣特征值、矩陣的譜和偶圖等基本概念。接著重點(diǎn)敘述了有向圖的特征值的主要結(jié)論及其證明，即是Perron-Frobenius定理、Levy-Desplanques定理、Gerschgorin圓盤定理、Brauer定理和Brualdi定理等，便于對(duì)有向圖的特征值的理解和掌握。最后描述了有向圖及其特征值在競(jìng)技比賽中的應(yīng)用，即是通過在單循環(huán)比賽中排列名次的實(shí)例，表明有向圖的特征值在實(shí)際應(yīng)用中的重要性。在研究有向圖的特征值的過程中，都要把有向圖化為矩陣的形式，再研究矩陣的特征值。對(duì)于高階矩陣，很難直接求出它們的特征值，于是，對(duì)有向圖的特征值的估值是1個(gè)重要的課題，本文對(duì)此進(jìn)行了研究。

關(guān)鍵字:有向圖；矩陣；特征值；圓盤；競(jìng)賽圖。

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.

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