Isomorphic gcd-graphs over polynomial rings

TL;DR

This paper explores gcd-graphs over polynomial rings with finite field coefficients, revealing their properties, isomorphism conditions, and spectral similarities, extending classical integer gcd-graph results to a polynomial setting.

Contribution

It introduces the study of gcd-graphs over polynomial rings, analyzes their properties, and investigates isomorphism and spectral conditions, expanding the understanding beyond integer gcd-graphs.

Findings

Gcd-graphs over polynomial rings share many properties with those over integers.

Two gcd-graphs over polynomial rings can be isomorphic more frequently than in the integer case.

The paper proposes analogs of conjectures related to gcd-graph isomorphisms and spectra.

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.