Biregular graph

Vertex counts

An ${\displaystyle (x,y)}$-biregular graph ${\displaystyle G=(U,V,E)}$ must satisfy the equation ${\displaystyle x|U|=y|V|}$. This follows from a simple double counting argument: the number of endpoints of edges in ${\displaystyle U}$ is ${\displaystyle x|U|}$, the number of endpoints of edges in ${\displaystyle V}$ is ${\displaystyle y|V|}$, and each edge contributes the same amount (one) to both numbers.

Symmetry

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

Configurations

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.