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On the gcd graphs over polynomial rings

Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân

TL;DR

This work extends gcd-graph theory to polynomial rings $\boldsymbol{F}_q[x]$, defining $G_f(D)$ as a Cayley graph on $R=\boldsymbol{F}_q[x]/(f)$ with generating set $S_D$ determined by divisors of $f$. It establishes a function-field–number-field correspondence, showing that gcd-graphs over $\boldsymbol{F}_q[x]$ inherit integral spectra described by Ramanujan sums, and develops a robust framework—via symmetric algebras and Ramanujan sums—for computing eigenvalues of $G_f(D)$. The paper provides deep graph-theoretic analyses: conditions for connectedness and anti-connectedness, a full bipartite classification, primeness criteria linked to homogeneous ideals, and a spectral formula guaranteeing integrality. It also investigates perfectness, gives sufficient conditions for non-perfection, and proves a powerful induced-subgraph universality result, showing that every finite graph is realizable as an induced subgraph of a gcd-graph. Collectively, these results deepen the analogy between gcd-graphs over $\mathbb{Z}$ and their function-field counterparts, and they broaden the toolkit for arithmetic graph constructions and their spectral properties in finite-field settings.

Abstract

Gcd-graphs over the ring of integers modulo $n$ are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over $\mathbb{Z}.$

On the gcd graphs over polynomial rings

TL;DR

This work extends gcd-graph theory to polynomial rings , defining as a Cayley graph on with generating set determined by divisors of . It establishes a function-field–number-field correspondence, showing that gcd-graphs over inherit integral spectra described by Ramanujan sums, and develops a robust framework—via symmetric algebras and Ramanujan sums—for computing eigenvalues of . The paper provides deep graph-theoretic analyses: conditions for connectedness and anti-connectedness, a full bipartite classification, primeness criteria linked to homogeneous ideals, and a spectral formula guaranteeing integrality. It also investigates perfectness, gives sufficient conditions for non-perfection, and proves a powerful induced-subgraph universality result, showing that every finite graph is realizable as an induced subgraph of a gcd-graph. Collectively, these results deepen the analogy between gcd-graphs over and their function-field counterparts, and they broaden the toolkit for arithmetic graph constructions and their spectral properties in finite-field settings.

Abstract

Gcd-graphs over the ring of integers modulo are a natural generalization of unitary Cayley graphs. The study of these graphs has foundations in various mathematical fields, including number theory, ring theory, and representation theory. Using the theory of Ramanujan sums, it is known that these gcd-graphs have integral spectra; i.e., all their eigenvalues are integers. In this work, inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We establish some fundamental properties of these graphs, emphasizing their analogy to their counterparts over
Paper Structure (11 sections, 45 theorems, 87 equations, 2 figures)

This paper contains 11 sections, 45 theorems, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background from graph theory
  3. Connectedness of gcd-graphs
  4. Bipartite property
  5. Primeness of gcd-graphs
  6. Spectrum of gcd-graphs
  7. Symmetric algebras
  8. Ramanujan sums over $\mathop{\mathrm{\mathbb{F}}}\nolimits_q[x]$
  9. Spectrum of $G_{f}(D)$
  10. Perfect gcd-graphs
  11. Induced subgraphs of gcd-graphs

Key Result

Theorem 1.1

An $\mathop{\mathrm{\mathbb{Z}}}\nolimits/n\mathop{\mathrm{\mathbb{Z}}}\nolimits$-circulant graph $G$ is an integral graph if and only if $G = G_n(D)$ for some $D.$

Figures (2)

  • Figure 1: The gcd-graph $G_{x(x+1)}(\{x, x+1 \})$
  • Figure 2: The gcd-Cayley graph $G_{x^2(x+1)}(\{x, x+1 \})$

Theorems & Definitions (99)

  • Theorem 1.1: See so2006integral
  • Remark 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 2.1: The complete graph $K_n$
  • Definition 2.2: Tensor product of graphs
  • Definition 2.3: Wreath product of graphs
  • Definition 2.4: Graph morphisms
  • Definition 2.5: Induced subgraphs
  • Lemma 3.1
  • ...and 89 more