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On gcd-graphs over finite rings

Tung T. Nguyen, Nguyen Duy Tân

TL;DR

This work extends gcd-graphs to finite rings, defining them via unit-stable generating sets and principal-ideal data, and studies connectivity, diameter, and spectrum in this broad setting. It provides an explicit generating-set description and a sharp connectivity-diameter framework that reduces to known results for $\mathbb{Z}/n$, while also proving integrality and giving a unified Ramanujan-sum-based spectrum formula for finite symmetric $\mathbb{Z}/n$-algebras. The results unify various gcd-graph constructions in the literature, offer new diameter bounds via reductions to cubelike graphs, and yield explicit eigenvalue formulas that are invariant under unit scaling, with potential implications for quantum information applications such as perfect state transfer. Overall, the paper lays a ring-theoretic foundation for gcd-graphs, bridging number-theoretic spectral techniques with finite-ring Cayley graphs.

Abstract

Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define and establish some foundational properties of gcd-graphs defined over a general finite commutative ring. In particular, we investigate the connectivity and diameter of these graphs. Additionally, when the ring is a finite symmetric $\mathbb{Z}/n$-algebra, we give an explicit description of their spectrum using the theory of Ramanujan sums that gives a unified treatment of various results in the literature.

On gcd-graphs over finite rings

TL;DR

This work extends gcd-graphs to finite rings, defining them via unit-stable generating sets and principal-ideal data, and studies connectivity, diameter, and spectrum in this broad setting. It provides an explicit generating-set description and a sharp connectivity-diameter framework that reduces to known results for , while also proving integrality and giving a unified Ramanujan-sum-based spectrum formula for finite symmetric -algebras. The results unify various gcd-graph constructions in the literature, offer new diameter bounds via reductions to cubelike graphs, and yield explicit eigenvalue formulas that are invariant under unit scaling, with potential implications for quantum information applications such as perfect state transfer. Overall, the paper lays a ring-theoretic foundation for gcd-graphs, bridging number-theoretic spectral techniques with finite-ring Cayley graphs.

Abstract

Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define and establish some foundational properties of gcd-graphs defined over a general finite commutative ring. In particular, we investigate the connectivity and diameter of these graphs. Additionally, when the ring is a finite symmetric -algebra, we give an explicit description of their spectrum using the theory of Ramanujan sums that gives a unified treatment of various results in the literature.

Paper Structure

This paper contains 5 sections, 20 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction and motivations
  2. Outline
  3. Gcd-graphs over a finite ring
  4. Connectivity of gcd-graphs
  5. Spectrum of $G_{R}(D).$

Key Result

Lemma 2.2

(See kaplansky1949elementary Let $R$ be an Artinian ring. If $Ra = Rb$, then there exists $u \in R^{\times}$ such that $b= ua.$

Figures (1)

  • Figure 1: The gcd-graph $G_{R}(D)$ where $R=\mathop{\mathrm{\mathbb{F}}}\nolimits_3[x]/x^2 \times \mathop{\mathrm{\mathbb{F}}}\nolimits_2$, $D = \{R(1,1), R(x,0)\}$

Theorems & Definitions (53)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Corollary 2.7
  • proof
  • Lemma 2.8
  • ...and 43 more