Biregular graph

In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph is a bipartite graph $$G=(U,V,E)$$ 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 $$U$$ is $$x$$ and the degree of the vertices in $$V$$ is $$y$$, then the graph is said to be $$(x,y)$$-biregular.



Example
Every complete bipartite graph $$K_{a,b}$$ is $$(b,a)$$-biregular. The rhombic dodecahedron is another example; it is (3,4)-biregular.

Vertex counts
An $$(x,y)$$-biregular graph $$G=(U,V,E)$$ must satisfy the equation $$x|U|=y|V|$$. This follows from a simple double counting argument: the number of endpoints of edges in $$U$$ is $$x|U|$$, the number of endpoints of edges in $$V$$ is $$y|V|$$, 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.