Version 1
: Received: 4 July 2024 / Approved: 5 July 2024 / Online: 5 July 2024 (07:28:33 CEST)
How to cite:
Gilevich, A.; Shoval, S.; Nosonovsky, M.; Frenkel, M.; Bormashenko, E. Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs. Preprints2024, 2024070494. https://doi.org/10.20944/preprints202407.0494.v1
Gilevich, A.; Shoval, S.; Nosonovsky, M.; Frenkel, M.; Bormashenko, E. Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs. Preprints 2024, 2024070494. https://doi.org/10.20944/preprints202407.0494.v1
Gilevich, A.; Shoval, S.; Nosonovsky, M.; Frenkel, M.; Bormashenko, E. Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs. Preprints2024, 2024070494. https://doi.org/10.20944/preprints202407.0494.v1
APA Style
Gilevich, A., Shoval, S., Nosonovsky, M., Frenkel, M., & Bormashenko, E. (2024). Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs. Preprints. https://doi.org/10.20944/preprints202407.0494.v1
Chicago/Turabian Style
Gilevich, A., Mark Frenkel and Edward Bormashenko. 2024 "Converting Tessellations into Graphs: from Voronoi Tessellations to Complete Graphs" Preprints. https://doi.org/10.20944/preprints202407.0494.v1
Abstract
A mathematical procedure enabling the transformation of an arbitrary tessellation of a surface into a bi-colored complete graph is introduced. Polygons constituting the tessellation are represented by vertices of the graphs. Vertices of the graphs are connected by two kinds of links/edges, namely, by a green link, when polygons have the same number of sides, and by a red link, when the polygons have a different number of sides. This procedure gives rise to a semi-transitive, complete, bi-colored Ramsey graph. The Ramsey number was established as R_trans (3,3)=5. Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits ζ=lim┬(N→∞)〖N_g/N_r 〗, where N is the total number of green and red seeds, N_g and N_r, were found ζ=0.272±0.001 (Voronoi) and ζ= 0.47±0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as S= 1.690±0.001 and for the Poisson line tessellation as S =1.265±0.015.
Computer Science and Mathematics, Discrete Mathematics and Combinatorics
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.
