2024/11/04 by Ján Mináč, Mináč, Ján, Tung T. Nguyen +4 · 1 citation
Computer Science · Mathematics · #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT) #Rings, Modules, and Algebras #math.NT
paper · pdf · doi:10.48550/arxiv.2411.01768
To appear in The Journal of the Australian Mathematical Society
openalex publication_date 2024/11/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28 · arxiv created 2026/08/03 · arxiv updated 2026/08/04
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 ℤ, it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.