A graph is edge-transitive (s-arc-transitive, respectively) if its full automorphism group Aut\, () acts transitively on the set of edges (the set of s-arcs in for an integer s 0, respectively). A 1-arc-transitive graph is called an arc-transitive graph or a symmetric graph. In this paper, we construct cubic symmetric bi-Cayley graphs over some groups of order p⁴, where p 7 is a prime. Using these constructions, we classify the connected cubic edge-transitive graphs of order 2p⁴ for each prime p and we also show that all these graphs are symmetric.
Wang et al. (Fri,) studied this question.