On $2$-arc-transitive graphs of order $kp^n$

TL;DR

This paper proves that for certain highly symmetric graphs of order involving a prime power, the prime is bounded, implying only finitely many such graphs exist under specified conditions, extending previous results.

Contribution

It establishes a bound on the prime order in $2$-arc-transitive graphs with large valency, generalizing recent findings by Conder, Li, and Potočnik.

Findings

Finiteness of $2$-arc-transitive graphs with prime power order and large valency

Existence of functions bounding the prime in terms of valency and parameters

Extension of previous finiteness results in symmetric graph theory

Abstract

We show that there exist functions $c$ and $g$ such that, if $k$, $n$ and $d$ are positive integers with $d> g(n)$ and $\Gamma$ is a $d$-valent $2$-arc-transitive graph of order $kp^n$ with $p$ a prime, then $p\leqslant kc(d)$. In other words, there are only finitely many $d$-valent 2-arc-transitive graphs of order $kp^n$ with $d>g(n)$ and $p$ prime. This generalises a recent result of Conder, Li and Poto\v{c}nik.