Biregular graph explained

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

K _{a,b} is (b,a) -biregular.The rhombic dodecahedron is another example; it is (3,4)-biregular.^[3]

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.^[4] 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]

Notes and References

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