双弧竞赛图及其得分向量  

On the Number of Double-arc Tournaments

作　　者：钱建国[1] 林晓霞[2] 

机构地区：[1]厦门大学数学科学学院,福建厦门361005 [2]集美大学数学系,福建厦门361021

出　　处：《厦门大学学报（自然科学版）》2004年第5期588-591,共4页Journal of Xiamen University：Natural Science

基　　金：福建省自然科学基金(F0210011)资助

摘　　要：双弧竞赛图是任意两点间恰有两条弧连接的有向图.本文旨在研究双弧竞赛图的得分向量,给出了非负整数向量(s1,s2,…,sn)是一个双弧竞赛图的得分向量的一个充分必要条件.刻画了具有固定得分向量的双弧竞赛图数目的生成函数并据此研究了得分向量计数的一些极值问题.最后,利用本原矩阵绝对值最大的特征值及有向图的性质讨论了选手的排名问题.The double-arc tournament is a digraph in which each pair of vertices are joined by two arcs. A double-arc tournament can be interpreated as a competition where n participants play each other just once and where, the ties are allowed.This paper aims at studying the properties of the score-vector of the double-arc tournament.A sufficiant and necessary condition for a non-negative integral vector (s_1,s_2,…,s_n)to be the score-vector of a double-arc tournament is given out. The generating function for the cardinalities T_2(S) of the double-arc tournaments with given score-vector S is introduced and studied, which may reveal some extremal properties of the number T_2(S).Finally, by using the techniques of the primitive matrix and the directed graph, the problem about ranking the participants is also studied.

关 键 词：得分向量 双弧竞赛图 有向图 本原矩阵 生成函数 充分必要条件 刻画 排名 最大 问题 

分 类 号：O157.5[理学—数学] O151.21[理学—基础数学]