| Diamond graph | |
| Vertices: | 4 |
| Edges: | 5 |
| Automorphisms: | 4 (Klein four-group: |
| Diameter: | 2 |
| Girth: | 3 |
| Radius: | 1 |
| 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] It consists of a complete graph 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,[2] 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.[3]
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 with itself.
The characteristic polynomial of the diamond graph is . It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.