Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degree

TL;DR

This paper investigates permutation groups with small minimal degree and classifies vertex-transitive graphs with motion 4 or prime, advancing understanding of graph symmetries and automorphism groups.

Contribution

It develops new results on primitive and imprimitive permutation groups with small minimal degree and classifies certain vertex-transitive graphs based on their motion.

Findings

Classified vertex-transitive graphs with motion 4 or prime.

Established properties of permutation groups with small minimal degree.

Connected small motion graphs to group-theoretic classifications.

Abstract

The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results we classify vertex-transitive graphs whose motion is $4$ or a prime number.