Cubic symmetric graphs of order a small number times a prime or a prime square

A graph is s-regular if its automorphism group acts regularly on the set of its s-arcs. In this paper, the s-regular elementary abelian coverings of the complete bipartite graph K"3","3 and the s-regular cyclic or elementary abelian coverings of the complete graph K"4 for each s>=1 are classified when the fibre-preserving automorphism groups act arc-transitively. A new infinite family of cubic 1-regular graphs with girth 12 is found, in which the smallest one has order 2058. As an interesting application, a complete list of pairwise non-isomorphic s-regular cubic graphs of order 4p, 6p, 4p^2 or 6p^2 is given for each s>=1 and each prime p.

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