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Isomorphic gcd-graphs over polynomial rings

Ján Mináč, Tung T. Nguyen, Nguyen Duy Tân

Abstract

Gcd-graphs over the ring of integers modulo $n$ are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent 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 discover that, in both cases, gcd-graphs share many similar and analogous properties. In this article, we extend this line of research further. Among other topics, we explore an analog of a conjecture of So and a weaker version of Sander-Sander, concerning the conditions under which two gcd-graphs are isomorphic or isospectral. We also provide several constructions showing that, unlike the case over $\mathbb{Z}$, it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.

Isomorphic gcd-graphs over polynomial rings

Abstract

Gcd-graphs over the ring of integers modulo are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent 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 discover that, in both cases, gcd-graphs share many similar and analogous properties. In this article, we extend this line of research further. Among other topics, we explore an analog of a conjecture of So and a weaker version of Sander-Sander, concerning the conditions under which two gcd-graphs are isomorphic or isospectral. We also provide several constructions showing that, unlike the case over , it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.

Paper Structure

This paper contains 13 sections, 27 theorems, 71 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Outline.
  3. Code
  4. Spectral vector determines $D$
  5. Gcd-graphs associated with prime powers
  6. Graph theoretic properties of $G_{P^k}(D)$
  7. Spectral properties of $G_{P^k}(D)$
  8. Congruence of eigenvalues of $G_{P^k}(D)$
  9. Some special eigenvalues of $G_{P^k}(D)$
  10. Some estimations on the eigenvalues of $G_{P^k}(D)$
  11. Isomorphic gcd-graphs
  12. Isomorphic unitary Cayley graphs in the family $\mathop{\mathrm{\mathbb{F}}}\nolimits_q[x]/f$
  13. Isomorphic gcd-graphs

Key Result

Theorem 2.3

Let $D_1, D_2$ be two proper subsets of $Div(f).$ Suppose that $G_{f}(D_1)$ and $G_{f}(D_2)$ have the same spectral vector. Then $D_1 = D_2.$

Theorems & Definitions (63)

  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 53 more