I know that, by D. McKay and C. E. Praeger papers" Vertextransitive graphs which are not Cayley graphs I", there exist 112 nonCayley vertextransitive graph with 24 vertices. Is there any such graph of valency 5? Which graph is the answer?
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Yes, there is one, see this paper.
You can fetch it from Gordon Royle's collection.

$\begingroup$ Dear Prof. McKay, thank you for your answer. I need the adjacency of vertices of this unique graph. I want to compute the automorphism group of this graph. Is the graph a wellknow graph? $\endgroup$ – majid arezoomand Nov 15 '15 at 14:41
