# Biregular graph

 Graph families defined by their automorphisms distance-transitive $\boldsymbol{\rightarrow}$ distance-regular $\boldsymbol{\leftarrow}$ strongly regular $\boldsymbol{\downarrow}$ symmetric (arc-transitive) $\boldsymbol{\leftarrow}$ t-transitive, t ≥ 2 skew-symmetric $\boldsymbol{\downarrow}$ (if connected) vertex- and edge-transitive $\boldsymbol{\rightarrow}$ edge-transitive and regular $\boldsymbol{\rightarrow}$ edge-transitive $\boldsymbol{\downarrow}$ $\boldsymbol{\downarrow}$ $\boldsymbol{\downarrow}$ vertex-transitive $\boldsymbol{\rightarrow}$ regular $\boldsymbol{\rightarrow}$ (if bipartite) biregular $\boldsymbol{\uparrow}$ Cayley graph $\boldsymbol{\leftarrow}$ zero-symmetric asymmetric

In graph-theoretic mathematics, a biregular graph[1] or semiregular bipartite graph[2] 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.

The graph of the rhombic dodecahedron is biregular.

## Example

Every complete bipartite graph $K_{a,b}$ is $(b,a)$-biregular.[3] The rhombic dodecahedron is another example; it is (3,4)-biregular.[4]

## 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.[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

1. ^ Scheinerman, Edward R.; Ullman, Daniel H. (1997), Fractional graph theory, Wiley-Interscience Series in Discrete Mathematics and Optimization, New York: John Wiley & Sons Inc., p. 137, ISBN 0-471-17864-0, MR 1481157.
2. ^ Dehmer, Matthias; Emmert-Streib, Frank (2009), Analysis of Complex Networks: From Biology to Linguistics, John Wiley & Sons, p. 149, ISBN 9783527627998.
3. ^ a b Lauri, Josef; Scapellato, Raffaele (2003), Topics in Graph Automorphisms and Reconstruction, London Mathematical Society Student Texts, Cambridge University Press, pp. 20–21, ISBN 9780521529037.
4. ^ Réti, Tamás (2012), "On the relationships between the first and second Zagreb indices" (PDF), MATCH Commun. Math. Comput. Chem. 68: 169–188.
5. ^ Gropp, Harald (2007), "VI.7 Configurations", in Colbourn, Charles J.; Dinitz, Jeffrey H., Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton) (Second ed.), Chapman & Hall/CRC, Boca Raton, FL, pp. 353–355.