TY - JOUR

T1 - Converting Tessellations into Graphs

T2 - From Voronoi Tessellations to Complete Graphs

AU - Gilevich, Artem

AU - Shoval, Shraga

AU - Nosonovsky, Michael

AU - Frenkel, Mark

AU - Bormashenko, Edward

N1 - Publisher Copyright:
© 2024 by the authors.

PY - 2024/8

Y1 - 2024/8

N2 - 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 semi-transitive number was established as (Formula presented.) Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits (Formula presented.), where N is the total number of green and red seeds, (Formula presented.) and (Formula presented.), were found (Formula presented.) 0.272 ± 0.001 (Voronoi) and (Formula presented.) 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as (Formula presented.) 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.

AB - 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 semi-transitive number was established as (Formula presented.) Shannon entropies of the tessellation and graphs are introduced. Ramsey graphs emerging from random Voronoi and Poisson Line tessellations were investigated. The limits (Formula presented.), where N is the total number of green and red seeds, (Formula presented.) and (Formula presented.), were found (Formula presented.) 0.272 ± 0.001 (Voronoi) and (Formula presented.) 0.47 ± 0.02 (Poisson Line). The Shannon Entropy for the random Voronoi tessellation was calculated as (Formula presented.) 1.690 ± 0.001 and for the Poisson line tessellation as S = 1.265 ± 0.015. The main contribution of the paper is the calculation of the Shannon entropy of the random point process and the establishment of the new bi-colored Ramsey graph on top of the tessellations.

KW - graph

KW - Ramsey theory

KW - random Voronoi diagram

KW - Shannon entropy

KW - tessellation

KW - topology

KW - transitivity

KW - Voronoi tessellation

UR - http://www.scopus.com/inward/record.url?scp=85200775581&partnerID=8YFLogxK

U2 - 10.3390/math12152426

DO - 10.3390/math12152426

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AN - SCOPUS:85200775581

SN - 2227-7390

VL - 12

JO - Mathematics

JF - Mathematics

IS - 15

M1 - 2426

ER -