Biregular graph

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

↑ 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 .
↑ 3.0 3.1 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", MATCH Commun. Math. Comput. Chem. 68: 169–188 .
↑ 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 .

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