On the number of generators of groups acting arc-transitively on graphs

TL;DR

This paper investigates bounds on the number of generators of groups acting arc-transitively on graphs, establishing that such bounds depend on graph size and group exponent, and showing they cannot depend solely on valency.

Contribution

It proves bounds on the number of vertices and group size based on graph and group parameters, and demonstrates the unboundedness of generators relative to valency alone.

Findings

Number of vertices and group size are bounded by functions of graph size and group exponent.

Number of generators cannot be bounded solely by the valency of the graph.

Provides new bounds relating group actions to graph parameters.

Abstract

Given a finite connected graph ${\Gamma}$ and a group $G$ acting transitively on the vertices of ${\Gamma}$, we prove that the number of vertices of ${\Gamma}$ and the cardinality of $G$ are bounded above by a function depending only on the cardinality of ${\Gamma}$ and on the exponent of $G$. We also prove that the number of generators of a group $G$ acting transitively on the arcs of a finite graph ${\Gamma}$ cannot be bounded by a function of the valency alone.