On Diophantine graphs

Abstract Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper, we study Diophantine graphs, that is, finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only if their product increased by one is a square. We provide various results for Diophantine graphs, including extendability properties, lower‐ and upper bounds for the maximum number of edges and chromatic numbers.