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 $$\varGamma $$ Γ is part of a well-understood family of graphs, or every non-identity automorphism of $$\varGamma $$ Γ fixes at most 1/3 of the edges. This answers a question proposed by Primož Potočnik and the third author.

Vertex-primitive digraphs with large fixity

“…To convey the flavour of his work, let us mention [2,Theorem 1.6], which states that the relative fixity of a strongly regular graph (other then a complete bipartite graph or the line graph of a complete graph) is at most 7 8 . The study of the fixity of graphs continued in a series of papers [4][5][6] by P. Spiga and coauthors (including the authors of the present paper), where the problem was studied in the context of vertex-transitive graphs of fixed valency.…”