摘 要
本文以图论的观点为主旨，对球队名次进行排序。把参赛球队的成绩看成是随机变量，建立随机模型。先在确定型的情况下，把参赛球队视为结点，用弧的指向表示比赛的胜负情况，通过直接或间接的方法构造竞赛图。在理论的指导下，寻找有向哈密顿路径或有向哈密顿回路，最终找出有向链，链头表示第1名，链尾表示最后1名，从而可以对球队进行排序。再在随机情况下，用极大似然法，结合各种参考细节，对球队获胜的概率进行估计，把估计所得的概率作为确定弧的权重的参考，根据权重结合确定型情况下球队的排序方法对随机比赛的球队进行排序，从而篮球队排序问题得到较好的解决。
关键词：竞赛图；有向哈密顿路径；哈密顿回路；极大似然估计
Abstract
This paper attempts to range the basketball teams with the theoretical basis of graph theory. The author takes the scores of the teams as random variables and establishes the random model. Under the condition of determinacy, we take the participating teams as the nodes and represent the results of the matches with arcs and directly or indirectly form the tournament graph. With the guiding of the theory, we first find out the directed Hamilton path or the Hamilton cycles and then find out the directed chains, the head of which represents the first place and the end of which the last place. Through this process we can ultimately range the team. Under the random condition, we can estimate the winning probabilities of the teams with the method of maximal plausibility combined with vary of concerned details. Then we take the estimated probabilities as the weightings of arcs. In the last place, we can range the random participating teams by taking into consideration the weightings combined with the method of ranging under the condition of determinacy. Through this way, the ranging problem of the basketball teams can be well settled.
Key words: tournament graph; directed Hamilton path; Hamilton cycles; maximal plausibility
前言
篮球运动是以投篮为中心的对抗性运动，篮球运动的复杂多变以及激烈对抗等特点，能够培养人的勇敢果断、积极进取的意志品质。篮球运动的问世，是球类游戏的高级发展，深受广大群众的喜爱，经常开展此项运动，对丰富业余文化生活，促进身心健康，提高工作和学习效率都起着积极作用。目前篮球比赛采用了比较科学的评分规则，1般采用积分制，但由于篮球比赛结果有较大的随机性且篮球比赛结果不具有传递性，因此研究篮球队排名次是1个10分有意义的问题。本文结合图论和概率论知识从另1个角度提出1种篮球队排名次的方法。在现有的排名规则下对篮球队排名进行1些合理的细化和设想。从而实现篮球队排名方式的可行性创新。具体做法是把球队比赛成绩看成随机变量，建立随机模型。从图论的观点出发，结点表示球队，带箭头的弧表示比赛的胜负，构造竞赛图，根据理论从竞赛图中找出有向链，从而得出确定型情况下球队的排名。再采用极大似然法得出球队随机情况下的名次，从而为篮球队排名次提出另1种比较科学的方法。
