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I know that, by D. McKay and C. E. Praeger papers" Vertex-transitive graphs which are not Cayley graphs I", there exist 112 non-Cayley vertex-transitive 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.

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  • $\begingroup$ 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 well-know graph? $\endgroup$
    – majid arezoomand
    Commented Nov 15, 2015 at 14:41
  • $\begingroup$ You can get the graph from this file. You can use software, e.g. sagemath, to find out the automorphism group and the adjacency lists of the graph. $\endgroup$
    – fidbc
    Commented Nov 15, 2015 at 18:50

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