Heptavalent Symmetric Graphs with Certain Conditions.

A graph Γ is said to be symmetric if its automorphism group Aut ⁡ (Γ) acts transitively on the arc set of Γ. We show that if Γ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group G of automorphisms, then either G is normal in Aut ⁡ (Γ) , or Aut ⁡ (Γ) contains a non-abelian simple normal subgroup T such that G ≤ T and (G , T) is explicitly given as one of 11 possible exceptional pairs of non-abelian simple groups. If G is arc-transitive, then G is always normal in Aut ⁡ (Γ) , and if G is regular on the vertices of Γ , then the number of possible exceptional pairs (G , T) is reduced to 5. [ABSTRACT FROM AUTHOR]