Classifying cubic symmetric graphs of order 8p or 8p2

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, we classify the s-regular elementary Abelian coverings of the three-dimensional hypercube for each s ≥ 1 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 19 208. As an application of the classification, all cubic symmetric graphs of order 8p or 8p2 are classified for each prime p, 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 4p, 4p2, 6p or 6p2 are classified and of Cheng and Oxley’s classification of symmetric graphs of order 2p, in Y. Cheng, J. Oxley [On weakly symmetric graphs of order twice a prime, J. Combin. Theory B 42 (1987) 196–211]. © 2004 Elsevier Ltd. All rights reserved.