Arc-regular cubic graphs of order four times an odd integer |
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Authors: | Marston D. E. Conder Yan-Quan Feng |
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Affiliation: | 1. Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, 1142, New Zealand 2. Mathematics, Beijing Jiaotong University, Beijing, 100044, P.R. China
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Abstract: | 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?CWalter 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,p 3) where p is an odd prime; the smallest of these has order?9828. |
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