Geodesic
Arc-regular cubic graphs of order four times an odd integer
Journal of Algebraic Combinatorics, Tome 36 (2012) no. 1, pp. 21-31.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $4 m$ 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 $PSL(2, p ^{3})$ where $p$ is an odd prime; the smallest of these has order 9828.
Classification : valent, one-regular, Cayley, graphs, for, dihedral, groups, from, the, work, of, Kwak, and, Oh, [19], and, Wang
Keywords: keywords arc-regular graph, one-regular graph, symmetric graph, Cayley graph 
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     title = {Arc-regular cubic graphs of order four times an odd integer},
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Conder, Marston D.E.; Feng, Yan-Quan. Arc-regular cubic graphs of order four times an odd integer. Journal of Algebraic Combinatorics, Tome 36 (2012) no. 1, pp. 21-31. http://geodesic.mathdoc.fr/item/JAC_2012__36_1_a7/