On quasiprimitive edge-transitive graphs of odd order and twice prime valency

A graph is edge-transitive if its automorphism group acts transitively on the edge set. In this paper, we investigate the automorphism groups of edge-transitive graphs of odd order and twice prime valency. Let Gamma be a connected graph of odd order and twice prime valency, and let G be a subgroup of the automorphism group of Gamma. In the case where G acts transitively on the edge set and quasiprimitively on the vertex set of Gamma, we prove that either G is almost simple, or G is a primitive group of affine type. If further G is an almost simple primitive group, then, with two exceptions, the socle of G acts transitively on the edge set of Gamma.