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A Cayley (di)graph Cay (G , S) of a group G is called normal if the right regular representation of G is normal in the full automorphism group of Cay (G , S) , and a CI-(di)graph if for every Cayley (di)graph Cay (G , T) , Cay (G , S) ≅ Cay (G , T) implies that there is σ ∈ Aut (G) such that S σ = T. We call a group G an NDCI-group or NCI-group if all normal Cayley digraphs or graphs of G are CI-digraphs or CI-graphs, respectively. We prove that a cyclic group of order n is an NDCI-group if and only if 8 ∤ n , and an NCI-group if and only if either n = 8 or 8 ∤ n. [ABSTRACT FROM AUTHOR]

Let G be a finite non-abelian simple group and let Γ be a connected tetravalent 2-arc-transitive G-regular graph. In 2004, Fang, Li, and Xu proved that either G is normal in the full automorphism group of Γ, or G is one of up to 22 exceptional candidates. In this paper, the number of exceptions is reduced to 7, and for each one, it is shown that has a normal arc-transitive non-abelian simple subgroup T such that and the pair (G, T) is explicitly given. Furthermore, there exists a G-regular -arc-transitive graph for each of the 7 pairs (G, T). [ABSTRACT FROM AUTHOR]

Abstract: A graph is -regular if its automorphism group acts regularly on the set of its -arcs. In this paper, we classify the -regular elementary Abelian coverings of the three-dimensional hypercube for each whose fibre-preserving automorphism subgroups act arc-transitively. This gives a new infinite family of cubic 1-regular graphs, in which the smallest one has order 19208. As an application of the classification, all cubic symmetric graphs of order or are classified for each prime , as a continuation of the first two authors’ work, in Y.-Q. Feng, J.H. Kwak [Cubic symmetric graphs of order a small number times a prime or a prime square (submitted for publication)] in which all cubic symmetric graphs of order , , or are classified and of Cheng and Oxley’s classification of symmetric graphs of order , in Y. Cheng, J. Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987) 196–211]. [Copyright &y& Elsevier]