An Isomorphism Problem in Z^2

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Theory and Applications of Graphs, ISSN: 2470-9859, Vol: 2, Issue: 1, Page: 1

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Noble, Matt
Georgia Southern University
Euclidean distance graph; graph isomorphisms; Discrete Mathematics and Combinatorics
article description
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.