Vertex-primitive digraphs with large fixity

Abstract

The relative fixity of a digraph $$\Gamma $$ Γ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $$\Gamma $$ Γ and the number of vertices of $$\Gamma $$ Γ . We characterize the vertex-primitive digraphs whose relative fixity is at least $$\frac{1}{3}$$ 1 3 , and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.