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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. LetΓ{\varGamma}    be a connected graph of odd order and twice prime valency, and let  G  be a subgroup of the automorphism group ofΓ{\varGamma}   . In the case where  G  acts transitively on the edge set and quasiprimitively on the vertex set ofΓ{\varGamma}   , 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 ofΓ{\varGamma}   .

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