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Almost all standard double covers of abelian Cayley graphs have smallest possible automorphism groups

Binzhou Xia, Zhishuo Zhang, Shasha Zheng

TL;DR

This work establishes a sharp asymptotic majority result for stability of Cayley graphs over finite abelian groups: as the group order $r$ grows, almost all inverse-closed connection sets $S$ yield Cayley graphs whose standard double cover has the smallest possible full automorphism group, namely $R(G)\rtimes\langle\iota\rangle$ extended by the inherent $C_2$ from the double cover. The authors implement a three-stage reduction to bound the problematic, unstable cases by decomposing the counting into structured subfamilies $\mathcal{S}_1$, $\mathcal{S}_2$, $\mathcal{S}_3$, $\mathcal{S}_4$, and $\mathcal{S}_5$, and derive explicit exponential-decay bounds for each. The main result yields that both labeled and unlabeled Cayley graphs on abelian groups are almost surely stable, implying that almost all finite abelian Cayley graphs are most rigid representations (MRRs). The methods connect stability criteria with the asymptotic enumeration framework developed in prior work, and the corollaries extend to refined statements for unlabeled graphs and automorphism-group exactness. Overall, the paper significantly advances understanding of stability in Cayley graphs by proving an essentially universal stability phenomenon in the abelian case.

Abstract

The standard double cover of a graph $Γ$ is the direct product $Γ\times K_2$. A graph $Γ$ is said to be stable if all the automorphisms of $Γ\times K_2$ come from its factors. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs of abelian groups, a complete classification remains elusive even for Cayley graphs of cyclic groups. In this paper, we study the asymptotic enumeration of both labeled and unlabeled Cayley graphs of abelian groups whose standard double cover has the smallest possible automorphism group. As a corollary, in both the labeled and unlabeled settings, we conclude that the proportion of stable Cayley graphs of an abelian group of order $r$ approaches $1$ as $r\rightarrow\infty$, proving that almost all Cayley graphs of finite abelian groups are stable.

Almost all standard double covers of abelian Cayley graphs have smallest possible automorphism groups

TL;DR

This work establishes a sharp asymptotic majority result for stability of Cayley graphs over finite abelian groups: as the group order grows, almost all inverse-closed connection sets yield Cayley graphs whose standard double cover has the smallest possible full automorphism group, namely extended by the inherent from the double cover. The authors implement a three-stage reduction to bound the problematic, unstable cases by decomposing the counting into structured subfamilies , , , , and , and derive explicit exponential-decay bounds for each. The main result yields that both labeled and unlabeled Cayley graphs on abelian groups are almost surely stable, implying that almost all finite abelian Cayley graphs are most rigid representations (MRRs). The methods connect stability criteria with the asymptotic enumeration framework developed in prior work, and the corollaries extend to refined statements for unlabeled graphs and automorphism-group exactness. Overall, the paper significantly advances understanding of stability in Cayley graphs by proving an essentially universal stability phenomenon in the abelian case.

Abstract

The standard double cover of a graph is the direct product . A graph is said to be stable if all the automorphisms of come from its factors. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs of abelian groups, a complete classification remains elusive even for Cayley graphs of cyclic groups. In this paper, we study the asymptotic enumeration of both labeled and unlabeled Cayley graphs of abelian groups whose standard double cover has the smallest possible automorphism group. As a corollary, in both the labeled and unlabeled settings, we conclude that the proportion of stable Cayley graphs of an abelian group of order approaches as , proving that almost all Cayley graphs of finite abelian groups are stable.
Paper Structure (12 sections, 30 theorems, 153 equations)

This paper contains 12 sections, 30 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction
  4. First reduction
  5. Second reduction
  6. Third reduction
  7. Proof of main results
  8. Bounding $|\mathcal{T}_\delta|$
  9. Bounding $|\mathcal{W}_\delta|$
  10. Bounding $|\mathcal{S}_5|$
  11. Proof of Theorem $\ref{['thmNumber']}$ and their corollaries
  12. Proof of Theorem $\ref{['thmIso']}$ and its corollaries

Key Result

Theorem 1.1

Let $G$ be an abelian group of order $r$ and of exponent greater than $2$, let $\iota$ be the inversion on $G$, and let $\delta\in(0,1/2)$. Then the proportion of inverse-closed subsets $S\subseteq G$ with is at least $1-h_\delta(r)$, where $h_\delta(r)$ is as in eqHdelta. In particular, the proportion of inverse-closed subsets $S\subseteq G$ such that $\mathrm{Cay}(G,S)$ is stable is at least $1

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Corollary 1.3: DSV2016
  • Theorem 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Definition 2.1: DX2000
  • Lemma 2.2
  • proof
  • ...and 53 more