Properties
The complete graph on n vertices is denoted by Kn. 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.
Kn has n(n − 1)/2 edges (a triangular number), and is a regular graph of degree n − 1. 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, ... (sequence A000085 in OEIS).
These numbers give the largest possible value of the Hosoya index for an n-vertex graph.
Read more about this topic: Complete Graph
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—John Locke (1632–1704)
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