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.
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.
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