# Diamond graph

Diamond graph
Vertices 4
Edges 5
Diameter 2
Girth 3
Automorphisms 4 (Z/2Z×Z/2Z)
Chromatic number 3
Chromatic index 3
Properties Hamiltonian
Planar
Unit distance

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.

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.

## Diamond-free graphs and forbidden minor

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.

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.

## Algebraic properties

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 Z/2Z 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.