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Asymptotic enumeration of Haar graphical representations

Yunsong Gan, Pablo Spiga, Binzhou Xia

TL;DR

The paper develops an extensive asymptotic theory for graphical representations arising from semiregular automorphism actions. By translating m-Cayley graphs into set-matrix data and employing a hierarchy of reductions (Babai–Godsil-like and Morris–Spiga-like) and a critical-pairs framework, the authors bound the number of Haar or m-Cayley graphs whose automorphism group exceeds the natural semiregular group. They perform a modular analysis across primitive group types (HS, HC, AS, PA, CD, HA, SD, TW) to show that, for large group orders, almost all Haar graphs are Haar graphical representations (HGRs) for nonabelian $G$, and a large majority of $m$-Cayley graphs/digraphs are graphical $m$-semiregular representations (G$m$SR/D$m$SR). The results yield significantly stronger Babai–Godsil-type bounds for DRRs/GRRs and provide asymptotic counts both with and without isomorphism identification, with extensions to $m$-partite, skew, and odd-quotient variants, thereby clarifying the prevalence of minimal automorphism groups in these graph classes. Overall, the work unifies group-theoretic and combinatorial enumeration techniques to resolve asymptotically the likelihood that semiregular automorphism structures yield minimal automorphism groups in large finite groups.

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.

Asymptotic enumeration of Haar graphical representations

TL;DR

The paper develops an extensive asymptotic theory for graphical representations arising from semiregular automorphism actions. By translating m-Cayley graphs into set-matrix data and employing a hierarchy of reductions (Babai–Godsil-like and Morris–Spiga-like) and a critical-pairs framework, the authors bound the number of Haar or m-Cayley graphs whose automorphism group exceeds the natural semiregular group. They perform a modular analysis across primitive group types (HS, HC, AS, PA, CD, HA, SD, TW) to show that, for large group orders, almost all Haar graphs are Haar graphical representations (HGRs) for nonabelian , and a large majority of -Cayley graphs/digraphs are graphical -semiregular representations (GSR/DSR). The results yield significantly stronger Babai–Godsil-type bounds for DRRs/GRRs and provide asymptotic counts both with and without isomorphism identification, with extensions to -partite, skew, and odd-quotient variants, thereby clarifying the prevalence of minimal automorphism groups in these graph classes. Overall, the work unifies group-theoretic and combinatorial enumeration techniques to resolve asymptotically the likelihood that semiregular automorphism structures yield minimal automorphism groups in large finite groups.

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.
Paper Structure (25 sections, 53 theorems, 198 equations)

This paper contains 25 sections, 53 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $m$-Cayley (di)graphs
  4. Haar graphs and odd-quotient graphs
  5. Simple arithmetic results
  6. Counting and group structure
  7. Strategy to prove Theorem \ref{['THM001']}
  8. Babai-Godsil-like reduction
  9. Morris-Spiga-like reduction
  10. Critical pairs
  11. Primitive permutation groups with a large regular subgroup
  12. Estimate $|\mathcal{Z}_{\textup{HS}}(G)\cup\mathcal{Z}_{\textup{HC}}(G)|$
  13. Estimate $|\mathcal{Z}_{\textup{AS}}(G)|$
  14. Estimate $|\mathcal{Z}_{\textup{PA}}(G)|$
  15. Estimate $|\mathcal{Z}_{\textup{CD}}(G)|$
  16. ...and 10 more sections

Key Result

Theorem 1.2

Let $\varepsilon\in\left(0,0.1\right]$, and let $n_\varepsilon$ be a positive integer such that for all $n\geq n_\varepsilon$, Let $G$ be a finite group of order $n$, and let $f_\varepsilon(n)=\frac{n^{0.5-\varepsilon}}{24(\log_2n)^{2.5}}-\frac{3\log_2^2n}{4}-15$.

Theorems & Definitions (98)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Morris-Spiga
  • Corollary 1.5
  • Theorem 1.6: Xia-Zheng
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Definition 2.3
  • Definition 2.4
  • ...and 88 more