Abstract
A graph is called arc-regular if its full automorphism group acts regularly on its arc set. We characterise arc-regular graphs with prime valency of cube-free order, and prove that such graphs of order twice an odd cube-free integer are normal Cayley graphs or binormal Cayley graphs, and there is no such graph of order four times an odd cube-free integer, which generalises certain previous results in the literature; we also classify arc- regular graphs with prime valency of order eight times an odd square-free integer, and found some new arc-regular graphs. Moreover, motivated by the obtained results, a conjecture and some problems are proposed.
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