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Combinatorial, geometric and probabilistic aspects of infinite sphere packings

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Názov: Combinatorial, geometric and probabilistic aspects of infinite sphere packings
Autori: Coderch Sendrós, Guillem
Prispievatelia: Universitat Politècnica de Catalunya. Departament de Matemàtiques, Rué Perna, Juan José
Informácie o vydavateľovi: Universitat Politècnica de Catalunya
Rok vydania: 2021
Zbierka: Universitat Politècnica de Catalunya, BarcelonaTech: UPCommons - Global access to UPC knowledge
Predmety: Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria, Discrete geometry, Circle packing, Fary-Wagner Theorem, Koebe-Andreev-Thurston Theorem, He-Schramm Theorem, Geometria discreta, Classificació AMS::52 Convex and discrete geometry::52C Discrete geometry
Popis: This thesis focuses on the Circle Packing Theorem for graphs representations and how to relate it with the combinatorial structure of this mathematical objects. We will study the Koebe-Andreev-Thurston Theorem, a very important result in contemporary combinatorics which states that every planar graph can be represented as a disk diagram (can be circle packed). Moreover, we will show the generalization for infinite planar graphs, the He-Schramm Theorem, that also relates the geometry of this circle packing in the plane with the internal composition of the graph. Electric networks and random walks will be very useful in the demonstration process of this last statement, so we also introduce them in this thesis.
Druh dokumentu: bachelor thesis
Popis súboru: application/pdf
Jazyk: English
Relation: http://hdl.handle.net/2117/349587
Dostupnosť: http://hdl.handle.net/2117/349587
Rights: http://creativecommons.org/licenses/by-nc-nd/3.0/es/ ; Open Access
Prístupové číslo: edsbas.C28C544
Databáza: BASE
Popis
Abstrakt:This thesis focuses on the Circle Packing Theorem for graphs representations and how to relate it with the combinatorial structure of this mathematical objects. We will study the Koebe-Andreev-Thurston Theorem, a very important result in contemporary combinatorics which states that every planar graph can be represented as a disk diagram (can be circle packed). Moreover, we will show the generalization for infinite planar graphs, the He-Schramm Theorem, that also relates the geometry of this circle packing in the plane with the internal composition of the graph. Electric networks and random walks will be very useful in the demonstration process of this last statement, so we also introduce them in this thesis.