Asymptotic enumeration of Haar graphical representations

TL;DR

This paper advances the enumeration of Haar graphical representations, showing that almost all Haar graphs of large nonabelian groups are HGRs, and improves bounds related to Cayley digraph automorphisms.

Contribution

It extends enumeration results to Haar graphical representations and proves their prevalence among Haar graphs of large nonabelian groups.

Findings

Proportion of HGRs among Haar graphs approaches 1 as group size increases.

Improved bounds on the proportion of DRRs among Cayley digraphs.

Enhanced understanding of automorphism groups in Cayley (di)graphs.

Abstract

This paper represents a significant leap forward in the problem of enumerating vertex-transitive graphs. Recent breakthroughs on symmetry of Cayley (di)graphs show that almost all finite Cayley (di)graphs have the smallest possible automorphism group. Extending the scope of these results, we enumerate (di)graphs admitting a fixed semiregular group of automorphisms with m orbits. Moreover, we consider the more intricate inquiry of prohibiting arcs within each orbit, where the special case m = 2 is known as the problem of finding Haar graphical representations (HGRs). We significantly advance the understanding of HGRs by proving that the proportion of HGRs among Haar graphs of a finite nonabelian group approaches 1 as the group order grows. As a corollary, we obtain an improved bound on the proportion of DRRs among Cayley digraphs in the solution of Morris and the second author to the Babai-Godsil conjecture.