Tetravalent half-transitive graphs of order

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 4 p are classified for each prime p . It is shown that there are no tetravalent half-transitive Cayley graphs of order 4 p and a tetravalent half-transitive non-Cayley graph of order 4 p exists if and only if p − 1 is divisible by 8, which is unique for a given order.