Finite edge-primitive graphs of prime valency

Many famous graphs are edge-primitive. Weiss (1973) and Guo et al. (2013) determined edge-primitive graphs of valency 3 and 5, respectively. In this paper, we study edge-primitive graphs of any prime valency. It is proved that all such graphs are 2-arc-transitive and the full automorphism groups are almost simple with the only exception the graphs being the complete bipartite graphs, and a complete classification is given of such graphs with soluble edge stabilizers, which widely extends the classifications of Weiss and Guo et al. (notice that the edge stabilizers of edge-primitive graphs with valency at most 5 are soluble).