An Isomorphism Problem in Z^2

Abstract

We consider Euclidean distance graphs with vertex set Q2 or Z2 and address the possibility or impossibility of finding isomorphisms between such graphs. It is observed that for any distances d1, d2 the non-trivial distance graphs G(Q2, d1) and G(Q2, d2) are isomorphic. Ultimately it is shown that for distinct primes p1, p2 the non-trivial distance graphs G(Z2, sqrt{p1}) and G(Z2, sqrt{p2}) are not isomorphic. We conclude with a few additional questions related to this work.

Keywords

• Euclidean distance graph
• graph isomorphisms