Biregular graph

In graph-theoretic mathematics, a biregular graph[1] or semiregular bipartite graph[2] is a bipartite graph

for which every two vertices on the same side of the given bipartition have the same degree as each other.

If the degree of the vertices in

and the degree of the vertices in

Every complete bipartite graph

[3] The rhombic dodecahedron is another example; it is (3,4)-biregular.

This follows from a simple double counting argument: the number of endpoints of edges in

, the number of endpoints of edges in

, and each edge contributes the same amount (one) to both numbers.

Every regular bipartite graph is also biregular.

Every edge-transitive graph (disallowing graphs with isolated vertices) that is not also vertex-transitive must be biregular.

[3] In particular every edge-transitive graph is either regular or biregular.

The Levi graphs of geometric configurations are biregular; a biregular graph is the Levi graph of an (abstract) configuration if and only if its girth is at least six.