The Edge-transitive Z_p×Z_p- Coverings Of Heawood Graph

Let graph X be a finite undirected simple graph. A graph is called edge-transitive, if AutX is transitive on the edge set E(X). A graph X is called a covering of X with a projection p : X → X, if p is a surjction from V(X) to V{X) such that P|N(v): N(v) →N(v) is a bijetion for any vertex v ∈ p^-1 (X) and v ∈ V(X) . The graph X is called the covering graph and X is the base. A covering X of X with a projection p is said to be regular (or K-covering) if there is a semiregular(on E{X) and V{X)) subgroup K of the automorphism group Aut(X) such that the graph X is isomophic to the quotient graph X/K, say by h, and the quotient mapX→X/K is the composition ph of p and h (for the purpose of this paper, all funtions are composed from left to right). In this paper we discuss The Edge-transitive Z_p × Z_p-coverings of Heawood graph, where p is a prime. we show that p = 7 or p (?) 1(mod7), and if the voltages {e,x₀} are linearly independent, and the voltages of the fundamental cycles satisfying x₀ = x₁ = x₂= x₅ = x₆, x₃ = x₄ = -e + x₀, then any such Z_p ×Z_p -covering is symmetric. As a result, we discuss that the Z_p×Z_P- coverings of Heawood graph are all 1-regular graphs, here p (?) 1(mod7), furthermore a new infinite family of cubic 1-regular graphs is constructed.