Biregular graph

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Graph families defined by their automorphisms
distance-transitivedistance-regularstrongly regular
symmetric (arc-transitive)t-transitive, t ≥ 2skew-symmetric
(if connected)
vertex- and edge-transitive
edge-transitive and regularedge-transitive
vertex-transitiveregular(if bipartite)
biregular
Cayley graphzero-symmetricasymmetric

In graph-theoretic mathematics, a biregular graph[1] or semiregular bipartite graph[2] is a bipartite graph 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 is and the degree of the vertices in is , then the graph is said to be -biregular.

The graph of the rhombic dodecahedron is biregular.

Example

Every complete bipartite graph is -biregular.[3] The rhombic dodecahedron is another example; it is (3,4)-biregular.[4]

Vertex counts

An -biregular graph must satisfy the equation . This follows from a simple double counting argument: the number of endpoints of edges in is , the number of endpoints of edges in is , 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]

References

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