## Abstract

Shannon entropy quantifying bi-colored Ramsey complete graphs is introduced and calculated for complete graphs containing up to six vertices. 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 (Formula presented.) -colored polygons with n α-sides or edges, and the fraction of monochromatic (Formula presented.) -colored convex polygons with m β-sides in the given complete graph. The introduced Shannon entropy is insensitive to the exact shape of the polygons, but it is sensitive to the distribution of monochromatic polygons in a given complete graph. The introduced Shannon entropies (Formula presented.) and (Formula presented.) are interpreted as follows: (Formula presented.) is interpreted as an average uncertainty to find the green (Formula presented.) polygon in the given graph; (Formula presented.) is, in turn, an average uncertainty to find the red (Formula presented.) polygon in the same graph. The re-shaping of the Ramsey theorem in terms of the Shannon entropy is suggested. Generalization for multi-colored complete graphs is proposed. Various measures quantifying the Shannon entropy of the entire complete bi-colored graphs are suggested. Physical interpretations of the suggested Shannon entropies are discussed.

Original language | English |
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Article number | 1427 |

Journal | Entropy |

Volume | 25 |

Issue number | 10 |

DOIs | |

State | Published - Oct 2023 |

## Keywords

- Ramsey number
- Ramsey theorem
- Shannon entropy
- Voronoi tessellation
- bi-colored graph
- complete graph