Abstract
In this paper, we study arc-transitive Cayley graphs on non-abelian simple groups with soluble stabilizers and valency seven. Let Γ be such a Cayley graph on a non-abelian simple group T. It is proved that either Γ is a normal Cayley graph or Γ is S-arc-transitive, with (S,T)=(An,An−1) and n=7,21,63 or 84; and, for each of these four values of n, there really exist some arc-transitive 7-valent non-normal Cayley graphs on An−1 and specific examples are constructed.
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