Tetravalent Connected Half-transitive Graphs of Order qp~2

A graph X is said to be a half-transitive graph if its full automorphism group denoted by Aut(X) acts transitively on its vertex set and edge set,but not on its arc set.In this paper,the connected half-transitive tetravalent graphs of order qp2(qp and are all odd primes) are proved to be isomorphic to a Normal Cayley graph of a metacyclic group,and the graph is also isomorphic to some tightly attached graph.For the automorphism of such graphs,its order and solvability are determined finally.