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Shannon Entropy of Ramsey Graphs
Version 1
: Received: 24 July 2023 / Approved: 24 July 2023 / Online: 25 July 2023 (10:27:56 CEST)
A peer-reviewed article of this Preprint also exists.
Frenkel, M.; Shoval, S.; Bormashenko, E. Shannon Entropy of Ramsey Graphs with up to Six Vertices. Entropy 2023, 25, 1427. Frenkel, M.; Shoval, S.; Bormashenko, E. Shannon Entropy of Ramsey Graphs with up to Six Vertices. Entropy 2023, 25, 1427.
Abstract
Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced. Complete graphs in which vertices are connected with two types of links, labeled as α-links and β-links are considered. Shannon-entropy is introduced according to the classical Shannon formula considering the fractions of monochromatic convex α-colored polygons with n α-sides or edges, and the fraction of monochromatic β-colored convex polygons with m β-sides in the given complete graph. Introduced Shannon entropy is insensitive to the exact shape of the graph, but it is sensitive to the distribution of monochromatic polygons in a given graph. The introduced Shannon Entropies Sα and Sβ are interpreted as follows: Sα is interpreted as an average uncertainty to find the green α-polygon in the given graph, Sβ is, in turn, an average uncertainty to find the red β-polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon Entropy is suggested. Various measures quantifying the Shannon Entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon Entropies are discussed.
Keywords
Complete graph; Shannon Entropy; bi-colored graph; Ramsey Theorem; Ramsey Number; Voronoi tessellation.
Subject
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.
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