On orders of automorphisms of vertex-transitive graphs

Abstract

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with n vertices and of valence d, d ≤ 4, is at most c d n where c 3 = 1 and c 4 = 9. Whether such a constant c d exists for valencies larger than 4 remains an unanswered question. Further, we prove that every automorphism g of a finite connected 3-valent vertex-transitive graph Γ, Γ ∼ = K 3,3 , has a regular orbit, that is, an orbit of g of length equal to the order of g. Moreover, we prove that in this case either Γ belongs to a well understood family of exceptional graphs or at least 5/12 of the vertices of Γ belong to a regular orbit of g. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms C of a connected 3-valent vertex-transitive graph Γ in terms of the number of vertices of Γ and the length of a longest orbit of C.

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

“…Investigations on the number of fixed points of graph automorphisms do have interesting applications. For instance, very recently Potočnik, Toledo and Verret [14] pivoting on the results in [11] have proved remarkable results on the cycle structure of general automorphisms of 3-valent vertex-transitive and 4-valent arc-transitive graphs.…”