An Isomorphism Problem in Z^2
oleh: Matt Noble
| Format: | Article |
|---|---|
| Diterbitkan: | Georgia Southern University 2015-01-01 |
Deskripsi
We consider Euclidean distance graphs with vertex set Q<sup>2</sup> or Z<sup>2</sup> and address the possibility or impossibility of finding isomorphisms between such graphs. It is observed that for any distances d<sub>1</sub>, d<sub>2</sub> the non-trivial distance graphs G(Q<sup>2</sup>, d<sub>1</sub>) and G(Q<sup>2</sup>, d<sub>2</sub>) are isomorphic. Ultimately it is shown that for distinct primes p<sub>1</sub>, p<sub>2</sub> the non-trivial distance graphs G(Z<sup>2</sup>, sqrt{p<sub>1</sub>}) and G(Z<sup>2</sup>, sqrt{p<sub>2</sub>}) are not isomorphic. We conclude with a few additional questions related to this work.