Stabilizers of consistent walks

TL;DR

This paper investigates conditions under which consistent walks in arc-transitive graphs have trivial stabilizers, linking the size of minimal generating sets of automorphism groups to graph valence.

Contribution

It provides sufficient conditions for consistent walks to have trivial stabilizers and relates the size of generating sets to the graph's valence.

Findings

Consistent walks with trivial stabilizers are characterized.

The size of minimal generating sets is bounded by the valence.

Conditions are established for automorphism groups in arc-transitive graphs.

Abstract

A walk of length $n$ in a graph is consistent if there exists an automorphism of the graph that maps the initial $n-1$ vertices to the final $n-1$ vertices of the walk. In this paper we find some sufficient conditions for a consistent walk in an arc-transitive graph to have a trivial pointwise stabilizer. We show that in that case, the size of the smallest generating set of the group is bounded by the valence of the graph.