Converting Tessellations into Graphs: From Voronoi Tessellations to Complete Graphs

Artem Gilevich, Shraga Shoval, Michael Nosonovsky, Mark Frenkel and Edward Bormashenko ()
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Artem Gilevich: Department of Chemical Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel
Shraga Shoval: Department of Industrial Engineering and Management, Faculty of Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel
Michael Nosonovsky: Department of Mechanical Engineering, University of Wisconsin-Milwaukee, Milwaukee, WI 53211, USA
Mark Frenkel: Department of Chemical Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel
Edward Bormashenko: Department of Chemical Engineering, Ariel University, P.O. Box 3, Ariel 407000, Israel

Mathematics, 2024, vol. 12, issue 15, 1-12

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 semi-transitive number was established as R t r a n s ( 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. 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.

Keywords: tessellation; graph; Ramsey theory; transitivity; Voronoi tessellation; random Voronoi diagram; Shannon entropy; topology (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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