The following problem is considered: if H is a semiregular abelian subgroup of a transitive permutation group G acting on a finite set X, find conditions for (non)existence of G invariant partitions of X. Conditions presented in this paper are derived by studying spectral properties of associated G invariant digraphs. As an essential tool, irreducible complex characters of H are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht et al. (Proc Camb Philos Soc 70:211–218, 1971) classifying edgetransitive generalized Petersen graphs, is given.
The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism.
In this paper, we determine the full automorphism groups of rose window graphs that are not edge-transitive. As the full automorphism groups of edge-transitive rose window graphs have been determined, this complete the problem of calculating the full automorphism group of rose window graphs. As a corollary, we determine which rose window graphs are vertex-transitive. Finally, we determine the isomorphism classes of non-edge-transitive rose window graphs.
A bicirculant is a graph admitting an automorphism with two cycles of equal length in its cycle decomposition. A graph is said to be arc-transitive if its automorphism group acts transitively on the set of its arcs. This paper gives a complete classification of connected pentavalent arc-transitive bicirculants.
The concept of generalized Cayley graphs was introduced by Marušič et al. (1992), where it was asked if there exists a vertex-transitive generalized Cayley graph which is not a Cayley graph. In this paper the question is answered in the affirmative with a construction of two infinite families of such graphs. It is also proven that every generalized Cayley graph admits a semiregular automorphism.