On the automorphism groups of graphs with 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 $\mathrm{\Gamma }$ be a connected graph of odd order and twice prime valency, and let $G$ be a subgroup of the automorphism group of $\text{Ga}$. In the case where $G$ acts transitively on the edges and quasiprimitively on the vertices of $\text{Ga}$, 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 edges of $\mathrm{\Gamma }$.