Tetravalent half-arc-transitive graphs of order

A graph is half-arc-transitive if its automorphism group acts transitively on vertices and edges, but not on arcs. It is known that for a prime p there is no tetravalent half-arc-transitive graphs of order p or p 2 . Xu [M.Y. Xu, Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275–282] classified the tetravalent half-arc-transitive graphs of order p 3 . As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p 4 . It shows that there are exactly p − 1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p 4 for each odd prime p .