Biregular graph - Alchetron, The Free Social Encyclopedia

→ {\displaystyle {\boldsymbol {\rightarrow }}} distance-regular ← {\displaystyle {\boldsymbol {\leftarrow }}} t-transitive, t ≥ 2 → {\displaystyle {\boldsymbol {\rightarrow }}} regular		← {\displaystyle {\boldsymbol {\leftarrow }}} strongly regular → {\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive

→ {\displaystyle {\boldsymbol {\rightarrow }}} edge-transitive and regular

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 a , b 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

Biregular graph Wikipedia

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Contents

Example

Vertex counts

Symmetry

Configurations

References