Isomorphic GCD-Graphs Over Polynomial Rings

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 (J. Mináč, T. T. Nguyen and N. D. Tân, ‘On the gcd-graphs over polynomial rings’, Canad. J. Math. , to appear. Published online (1 October 2025)), 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 and 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 Z double struck upper Z , it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.