Arc-regular cubic graphs of order four times an odd integer
نویسندگان
چکیده
A graph is arc-regular if its automorphism group acts sharply-transitively on the set of its ordered edges. This paper answers an open question about the existence of arc-regular 3-valent graphs of order 4m where m is an odd integer. Using the Gorenstein–Walter theorem, it is shown that any such graph must be a normal cover of a base graph, where the base graph has an arc-regular group of automorphisms that is isomorphic to a subgroup of Aut(PSL(2, q)) containing PSL(2, q) for some odd prime-power q. Also a construction is given for infinitely many such graphs— namely a family of Cayley graphs for the groups PSL(2,p3) where p is an odd prime; the smallest of these has order 9828.
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