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 $K_$ 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..

References

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Bipartite graphs