Asymptotic enumeration of graphical regular representations

TL;DR

This paper estimates the number of graphical regular representations (GRRs) of large groups and shows that almost all Cayley graphs have minimal automorphism groups, confirming two longstanding conjectures in the field.

Contribution

It provides asymptotic estimates for the number of GRRs and proves that almost all Cayley graphs have the smallest possible automorphism groups, confirming two key conjectures.

Findings

Almost all Cayley graphs have minimal automorphism groups

Confirmed the Babai-Godsil-Imrich-Lovasz conjecture on GRRs

Confirmed Xu's conjecture on normal Cayley graphs

Abstract

We estimate the number of graphical regular representations (GRRs) of a given group with large enough order. As a consequence, we show that almost all finite Cayley graphs have full automorphism groups 'as small as possible'. This confirms a conjecture of Babai-Godsil-Imrich-Lovasz on the proportion of GRRs, as well as a conjecture of Xu on the proportion of normal Cayley graphs, among Cayley graphs of a given finite group.