complete graph

Also known as complete digraph, complete graphs, complete digraphs, 2K1-free graph

simple undirected graph in which every pair of distinct vertices is connected by a unique edge

Key facts

Vertices: n
Edges: n ( n − 1 ) 2 {\displaystyle \textstyle {\frac {n(n-1)}{2}}}
Radius: { 0 n ≤ 1 1 otherwise {\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}
Diameter: { 0 n ≤ 1 1 otherwise {\displaystyle \left\{{\begin{array}{ll}0&n\leq 1\\1&{\text{otherwise}}\end{array}}\right.}
Girth: { ∞ n ≤ 2 3 otherwise {\displaystyle \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: { ∅ n = 0 { 0 1 } n = 1 { ( n − 1 ) 1 , − 1 n − 1 } otherwise {\displaystyle \left\{{\begin{array}{lll}\emptyset &n=0\\\left\{0^{1}\right\}&n=1\\\left\{(n-1)^{1},-1^{n-1}\right\}&{\text{otherwise}}\end{array}}\right.}
Properties: ( n − 1) -regular Symmetric graph Vertex-transitive Edge-transitive Strongly regular Integral
Notation: K n

via Wikipedia infobox

Wikidata facts

Subclass of: Moore graph
Image: Complete graph example.svg

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maintained by WikiProject: WikiProject Mathematics
Commons category: Complete graphs
graph diameter: 1
graph radius: 1
studied by: graph theory
opposite of: edgeless graph

Sources (2)

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Encyclopedic overview

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, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose.

Excerpted from Wikipedia’s “complete graph” article, available under the CC BY-SA 4.0 licence.