Erdős–Diophantine graph

An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any additional points.

Erdős–Diophantine graphs are named after Paul Erdős and Diophantus of Alexandria.

They form a subset of the set of Diophantine figures, which are defined as complete graphs in the Diophantine plane for which the length of all edges are integers (unit distance graphs).

By numerical search, Kohnert & Kurz (2007) have shown that three-node Erdős–Diophantine graphs do exist.

More generally, the total length of any closed path in an Erdős–Diophantine graph is always even.

A 5-node Erdős–Diophantine graph (node-distances as indicated).