Tetravalent half-arc-transitive graphs of order p4p4

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 pp there is no tetravalent half-arc-transitive graphs of order pp or p2p2. 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 p3p3. As a continuation, we classify in this paper the tetravalent half-arc-transitive graphs of order p4p4. It shows that there are exactly p−1p−1 nonisomorphic connected tetravalent half-arc-transitive graphs of order p4p4 for each odd prime pp.