Biregular graph

In graph-theoretic mathematics, a biregular graph^[1] or semiregular bipartite graph^[2] is a bipartite graph ${\displaystyle G=(U,V,E)}$ 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 ${\displaystyle U}$ is ${\displaystyle x}$ and the degree of the vertices in ${\displaystyle V}$ is ${\displaystyle y}$, then the graph is said to be ${\displaystyle (x,y)}$-biregular.

The graph of the rhombic dodecahedron is biregular.

Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, t ≥ 2		skew-symmetric
↓
(if connected) vertex- and edge-transitive	→	edge-transitive and regular	→	edge-transitive
↓		↓		↓
vertex-transitive	→	regular	→	(if bipartite) biregular
↑
Cayley graph	←	zero-symmetric		asymmetric

Example

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

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.^[3] 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.^[5]