In the mathematical field of graph theory, the diamond graph is a planar, undirected graph with 4 vertices and 5 edges.^[1][2] It consists of a complete graph ${\displaystyle K_{4}}$ minus one edge.

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Diamond graph
Vertices	4
Edges	5
Radius	1
Diameter	2
Girth	3
Automorphisms	4 (Klein four-group: ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} )}$
Chromatic number	3
Chromatic index	3
Properties	Hamiltonian Planar Unit distance
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The diamond graph has radius 1, diameter 2, girth 3, chromatic number 3 and chromatic index 3. It is also a 2-vertex-connected and a 2-edge-connected, graceful,^[3] Hamiltonian graph.

A graph is diamond-free if it has no diamond as an induced subgraph. The triangle-free graphs are diamond-free graphs, since every diamond contains a triangle. The diamond-free graphs are locally clustered: that is, they are the graphs in which every neighborhood is a cluster graph. Alternatively, a graph is diamond-free if and only if every pair of maximal cliques in the graph shares at most one vertex.

The family of graphs in which each connected component is a cactus graph is downwardly closed under graph minor operations. This graph family may be characterized by a single forbidden minor. This minor is the diamond graph.^[4]

If both the butterfly graph and the diamond graph are forbidden minors, the family of graphs obtained is the family of pseudoforests.

The full automorphism group of the diamond graph is a group of order 4 isomorphic to the Klein four-group, the direct product of the cyclic group ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ with itself.

The characteristic polynomial of the diamond graph is ${\displaystyle x(x+1)(x^{2}-x-4)}$. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.

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