Biregular graph

In graph-theoretic mathematics, a biregular graph[1] or semiregular bipartite graph[2] is a bipartite graph ${\displaystyle G=(U,V,E)}$ 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 ${\displaystyle U}$ is ${\displaystyle x}$ and the degree of the vertices in ${\displaystyle V}$ is ${\displaystyle y}$, then the graph is said to be ${\displaystyle (x,y)}$-biregular.

The graph of the rhombic dodecahedron is biregular.

Example

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

Vertex counts

An ${\displaystyle (x,y)}$-biregular graph ${\displaystyle G=(U,V,E)}$ must satisfy the equation ${\displaystyle x|U|=y|V|}$. This follows from a simple double counting argument: the number of endpoints of edges in ${\displaystyle U}$ is ${\displaystyle x|U|}$, the number of endpoints of edges in ${\displaystyle V}$ is ${\displaystyle y|V|}$, 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, archived from the original (PDF) on 2017-08-29, retrieved 2012-09-02.
5. ^ Gropp, Harald (2007), "VI.7 Configurations", in Colbourn, Charles J.; Dinitz, Jeffrey H. (eds.), Handbook of combinatorial designs, Discrete Mathematics and its Applications (Boca Raton) (Second ed.), Chapman & Hall/CRC, Boca Raton, Florida, pp. 353–355.