Vertex-primitive digraphs with large fixity

Abstract

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