Complete graph

Complete graph
$K 7$ , a complete graph with 7 vertices
Vertices	$n$
Edges	$\textstyle {\frac {n(n-1)}{2}}$ $\textstyle {\frac {n(n-1)}{2}}$
Radius	$\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$ $\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$
Diameter	$\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$ $\left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.$
Girth	$\left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.$ $\left\{{\begin{array}{ll}\infty &n\leq 2\\3&{\text{otherwise}}\end{array}}\right.$
Automorphisms	$n! (S n)$
Chromatic number	$n$
Chromatic index	$n$ if $n$ is odd $n - 1$ if $n$ is even
Spectrum	$\left\{{\begin{array}{lll}\emptyset &n=0\\\{0^{1}\}&n=1\\\{(n-1)^{1},-1^{n-1}\}&{\text{otherwise}}\end{array}}\right.$ $\left\{{\begin{array}{lll}\emptyset &n=0\\\{0^{1}\}&n=1\\\{(n-1)^{1},-1^{n-1}\}&{\text{otherwise}}\end{array}}\right.$
Properties	$(n - 1)$ -regular Symmetric graph Vertex-transitive Edge-transitive Strongly regular Integral
Notation	$K n$

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.

The complete graph on $n$ vertices is denoted by $K n$ . 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 $n (n - 1)/2$ 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.

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