On the orders of arc-transitive graphs

TL;DR

This paper investigates the possible sizes of arc-transitive graphs with given valency, establishing finiteness results for certain orders related to prime numbers and highlighting differences between 3-valent and higher valency cases.

Contribution

It provides new finiteness theorems for the orders of arc-transitive graphs with specified valency and prime-related sizes, advancing understanding of their structural limitations.

Findings

Finitely many 3-valent 2-arc-transitive graphs with order kp for prime p.

Finitely many d-valent 2-arc-transitive graphs with order kp or kp^2 for d ≥ 4.

Infinitely many k with only finitely many 3-valent symmetric graphs of order kp.

Abstract

A graph is called {\em arc-transitive} (or {\em symmetric}) if its automorphism group has a single orbit on ordered pairs of adjacent vertices, and 2-arc-transitive its automorphism group has a single orbit on ordered paths of length 2. In this paper we consider the orders of such graphs, for given valency. We prove that for any given positive integer $k$, there exist only finitely many connected 3-valent 2-arc-transitive graphs whose order is $kp$ for some prime $p$, and that if $d\ge 4$, then there exist only finitely many connected $d$-valent 2-arc-transitive graphs whose order is $kp$ or $kp^2$ for some prime $p$. We also prove that there are infinitely many (even) values of $k$ for which there are only finitely many connected 3-valent symmetric graphs of order $kp$ where $p$ is prime.