Primitive half-transitive graphs constructed from the symmetric groups of prime degrees

A graph is said to be half-transitive if its automorphism group acts transitively on the vertex set and edge set but intransitively on the arc set. In this paper, we construct infinitely many primitive half-transitive graphs with automorphism groups being the symmetric groups of prime degrees, and show that there exists at least one primitive half-transitive graph of valency 2p for a prime p no less than 7 and p ≠ 13 . As a byproduct of our construction, infinitely many primitive 2-arc-regular Cayley graphs are given.