In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or n-fan) Fn is a planar, undirected graph with 2n + 1 vertices and 3n edges.

[1] The friendship graph Fn can be constructed by joining n copies of the cycle graph C3 with a common vertex, which becomes a universal vertex for the graph.

[2] By construction, the friendship graph Fn is isomorphic to the windmill graph Wd(3, n).

It is unit distance with girth 3, diameter 2 and radius 1.

The friendship theorem of Paul Erdős, Alfréd Rényi, and Vera T. Sós (1966)[3] states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs.

Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others.

[4] A combinatorial proof of the friendship theorem was given by Mertzios and Unger.

[6] A formalised proof in Metamath was reported by Alexander van der Vekens in October 2018 on the Metamath mailing list.

Its chromatic polynomial can be deduced from the chromatic polynomial of the cycle graph C3 and is equal to The friendship graph Fn is edge-graceful if and only if n is odd.

According to extremal graph theory, every graph with sufficiently many edges (relative to its number of vertices) must contain a

These bounds generalize Turán's theorem on the number of edges in a triangle-free graph, and they are the best possible bounds for this problem (when

), in that for any smaller number of edges there exist graphs that do not contain a

[10] Any two vertices having exactly one neighbor in common is equivalent to any two vertices being connected by exactly one path of length two.

-graphs, in which any two vertices are connected by a unique path of length

no such graphs are known, and the claim of their non-existence is Kotzig's conjecture.