Arc-transitive elementary abelian covers of the Pappus graph

In this paper, all pairwise non-isomorphic p  -elementary abelian covering projections admitting a lift of an arc-transitive subgroup of the full automorphism group of the Pappus graph F18F18, the unique connected cubic symmetric graph of order 18, are constructed. The number of such covering projections is equal to 5, 19, 9, 11 and 5 if p=2p=2, p=3p=3, p≡7(mod12), p≡1(mod12) and p≡5(mod6), respectively. As results, three infinite families of cubic ss-regular graphs for s=1s=1, 2 and 3 are constructed, and a classification of the cubic ss-regular graphs of order 18p18p for each s≥1s≥1 and each prime pp is given. From the classification of cubic ss-regular graphs of order 18p18p we have the following: (1) apart from the two 5-regular graphs F90F90 and F234BF234B and the 2-regular graph F54F54, all of these graphs are 1-regular with p≡1(mod6); (2) apart from F90F90 and F234BF234B all of these graphs are of girth 6; (3) apart from F234BF234B all of these graphs are bipartite.