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Biregular graph

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]

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.

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 graphsofgeometric 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]