Heptavalent Symmetric Graphs with Certain Conditions

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