On the number of fixed edges of automorphisms of vertex-transitive graphs of small valency

Abstract We prove that, if $$\varGamma $$ Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either part of well-understood family graphs, every non-identity automorphism fixes at most 1/3 the edges. This answers question proposed by Primož Potočnik third author.