Article
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Embeddings of Graphs. Tessellate and Decussate Structures.
Version 1
: Received: 2 October 2023 / Approved: 3 October 2023 / Online: 4 October 2023 (11:21:00 CEST)
How to cite: O'Keeffe, M.; Treacy, M. M. J. Embeddings of Graphs. Tessellate and Decussate Structures.. Preprints 2023, 2023100188. https://doi.org/10.20944/preprints202310.0188.v1 O'Keeffe, M.; Treacy, M. M. J. Embeddings of Graphs. Tessellate and Decussate Structures.. Preprints 2023, 2023100188. https://doi.org/10.20944/preprints202310.0188.v1
Abstract
We address the problem of finding a unique graph embedding that best describes a graph's "topology". This issue is of particular interest in the chemistry of materials. Graphs that admit a tiling in 3-dimensional Euclidean space are termed tessellate, those that do not decussate. We give examples of decussate and tessellate graphs that are finite and 3-periodic. We conjecture that a graph has at most one tessellate embedding. We give reasons for considering this the default "topology" of periodic graphs.
Keywords
Graph; topology; ambient isotopy; periodic net; tessellate; decussate
Subject
Chemistry and Materials Science, Materials Science and Technology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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