Tetravalent half-transitive graphs of order 4p4p

A graph is half-transitive   if its automorphism group acts transitively on its vertex set and edge set, but not on its arc set. In this paper, the tetravalent half-transitive graphs of order 4p4p are classified for each prime pp. It is shown that there are no tetravalent half-transitive Cayley graphs of order 4p4p and a tetravalent half-transitive non-Cayley graph of order 4p4p exists if and only if p−1p−1 is divisible by 8, which is unique for a given order.