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 edge-transitive generalized Petersen graphs, is given.

COBISS.SI-ID: 1536772036

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.

COBISS.SI-ID: 16959577

Let S be a subset of the cyclic group Z_n. The cyclic Haar graph H(Z_n, S) is the bipartite graph with color classes Z_n^+ and Z_n^-, and edges {x^+, y^-}, where x, y belong to Z_n and y-x belongs to S. In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4.

COBISS.SI-ID: 1024507988