摘 要
本文以圖論的觀點為主旨，對球隊名次進行排序。把參賽球隊的成績看成是隨機變數，建立隨機模型。先在確定型的情況下，把參賽球隊視為結點，用弧的指向表示比賽的勝負情況，通過直接或間接的方法構造競賽圖。在理論的指導下，尋找有向哈密頓路徑或有向哈密頓迴路，最終找出有向鏈，鏈頭表示第1名，鏈尾表示最後1名，從而可以對球隊進行排序。再在隨機情況下，用極大似然法，結合各種參考細節，對球隊獲勝的概率進行估計，把估計所得的概率作為確定弧的權重的參考，根據權重結合確定型情況下球隊的排序方法對隨機比賽的球隊進行排序，從而籃球隊排序問題得到較好的解決。
關鍵詞：競賽圖；有向哈密頓路徑；哈密頓迴路；極大似然估計
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
籃球運動是以投籃為中心的對抗性運動，籃球運動的複雜多變以及激烈對抗等特點，能夠培養人的勇敢果斷、積極進取的意志品質。籃球運動的問世，是球類遊戲的高階發展，深受廣大群眾的喜愛，經常開展此項運動，對豐富業餘文化生活，促進身心健康，提高工作和學習效率都起著積極作用。目前籃球比賽採用了比較科學的評分規則，1般採用積分制，但由於籃球比賽結果有較大的隨機性且籃球比賽結果不具有傳遞性，因此研究籃球隊排名次是1個10分有意義的問題。本文結合圖論和概率論知識從另1個角度提出1種籃球隊排名次的`方法。在現有的排名規則下對籃球隊排名進行1些合理的細化和設想。從而實現籃球隊排名方式的可行性創新。具體做法是把球隊比賽成績看成隨機變數，建立隨機模型。從圖論的觀點出發，結點表示球隊，帶箭頭的弧表示比賽的勝負，構造競賽圖，根據理論從競賽圖中找出有向鏈，從而得出確定型情況下球隊的排名。再採用極大似然法得出球隊隨機情況下的名次，從而為籃球隊排名次提出另1種比較科學的方法。
