Asymmetrische reguläre Graphen

A classification of the graphical m-semiregular representation of finite groups

“…[11, Theorem 1] If X is a finite graph, then either X or X c is prime with respect to Cartesian multiplication unless X is one of the following six graphs: Proof. Let G be a cyclic group of order n. When n = 1, G admits an m-GRR if and only if there exists a regular asymmetric graph of order m. By a nice result of Baron and Imrich [4], we have m ≥ 10: there exist 4-regular asymmetric graphs of order m for each m ≥ 10, and a regular graph with fewer than 10 vertices is not asymmetric unless m = 1. Suppose n = 2.…”

On the number of outer automorphisms of the automorphism group of a right-angled Artin group

“…In fact, it is easily checked that the Frucht graph is austere. Baron-Imrich [1] generalised the Frucht graph to produce a family of finite, 3-regular graphs with trivial symmetry groups, over which n = |V | is unbounded. Like the Frucht graph, these graphs may also be shown to be austere, and so they define a class of right-angled Artin groups which proves Theorem B.…”

Godsil-McKay switching and isomorphism

“…. , ℓ}, x has either 0, 1 2 |Xi| or |Xi| neighbors in Xi. Moreover, suppose that for all i, j ∈ {1, .…”

“…. , ℓ} such that x has 1 2 |Xi| neighbors in Xi delete the corresponding 1 2 |Xi| edges and join x instead to the 1 2 |Xi| other vertices in Xi. Then G and G ′ are cospectral.…”