Semisymmetry of Generalized Folkman Graphs

A regular edge- but not vertex-transitive graph is said to be semisymmetric. The study of semisymmetric graphs was initiated by Folkman, who, among others, gave constructions of several infinite families such graphs . In this paper a generalization of his construction for orders a multiple of 4 is proposed, giving rise to some new families of semisymmetric graphs. In particular, one associated with the cyclic group of order n, n ≥ 5, which belongs to the class of tetracirculants, that is, graphs admitting an automorphism with precisely four orbits, all of the same length. Semisymmetry properties of tetracirculants are investigated in greater detail, leading to a classification of all semisymmetric graphs of order 4 p, where p is a prime.