Biregular graph

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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
\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 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

  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, 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 .