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Concept
Biregular graph
Summary
In graph-theoretic mathematics, a biregular graph or semiregular bipartite graph 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.
Every complete bipartite graph is -biregular.
The rhombic dodecahedron is another example; it is (3,4)-biregular.
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.
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.
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.
Official source
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