Biregular graph

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

^ 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.
^ Dehmer, Matthias; Emmert-Streib, Frank (2009), Analysis of Complex Networks: From Biology to Linguistics, John Wiley & Sons, p. 149, ISBN 9783527627998.
^ ^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.
^ 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.
^ 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.

[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.

[ls03-3] 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.

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Graph families defined by their automorphisms
distance-transitive	→	distance-regular	←	strongly regular
↓
symmetric (arc-transitive)	←	t-transitive, $t \geq 2$		skew-symmetric
↓
_{(if connected)} vertex- and edge-transitive	→	edge-transitive and regular	→	edge-transitive
↓		↓		↓
vertex-transitive	→	regular	→	_{(if bipartite)} biregular
↑
Cayley graph	←	zero-symmetric		asymmetric