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.

• Vámos matroid