Flag complex

In mathematics , flag complexes are certain simplicial complexes that play a role in graph theory , geometric topology and geometric group theory .

A flag complex.

definition

A flag complex is an abstract simplicial complex that fulfills the following condition (" Gromov's no -condition"): if there are a set of corners such that every two corners belong to a common simplex, then the corners of form a simplex. ${\displaystyle \Delta }$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle S}$

Examples

A graph is a complex of flags if and only if holds. Here refers to the length of a shortest circuit in . ${\displaystyle \Gamma }$ ${\displaystyle girth(\Gamma )\geq 4}$ ${\displaystyle girth(\Gamma )}$ ${\displaystyle \Gamma }$

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  Petersen graph : ${\displaystyle girth(\Gamma )=5}$

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  Heawood graph : ${\displaystyle girth(\Gamma )=6}$

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  McGee graph ${\displaystyle girth(\Gamma )=7}$

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  Tutte – Coxeter graph ${\displaystyle girth(\Gamma )=8}$

literature

• Daniel Wise: From Riches to Raags: 3-Manifolds, Right-Angled Artin Groups, and Cubical Geometry , CBMS Regional Conference Series in Mathematics 2012; 141 pp; softcover, ISBN 0-8218-8800-5