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In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, appeared already in the 13th century, in the work of Ramon Llull.〔. 〕 Such a drawing is sometimes referred to as a mystic rose.〔.〕 ==Properties== The complete graph on vertices is denoted by . Some sources claim that the letter K in this notation stands for the German word ''komplett'',〔.〕 but the German name for a complete graph, ''vollständiger Graph'', does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory.〔.〕 ''K''''n'' has edges (a triangular number), and is a regular graph of degree . All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. The number of matchings of the complete graphs are given by the telephone numbers :1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, 568504, 2390480, 10349536, 46206736, ... . These numbers give the largest possible value of the Hosoya index for an ''n''-vertex graph.〔.〕 The number of perfect matchings of the complete graph ''K''''n'' (with ''n'' even) is given by the double factorial (''n'' − 1)!!.〔.〕 The crossing numbers up to ''K''''27'' are known, with ''K''''28'' requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.〔(【引用サイトリンク】 Rectilinear Crossing Number project )〕 Crossing numbers for ''K''''n'' are :0, 0, 0, 0, 1, 3, 9, 19, 36, 62, 102, 153, 229, 324, 447, 603, 798, 1029, 1318, 1657, 2055, 2528, 3077, 3699, 4430, 5250, 6180, ... . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complete graph」の詳細全文を読む スポンサード リンク
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