Flag Complex
In an abstract simplicial complex, a set S of vertices that is not itself part of the complex, but such that each pair of vertices in S belongs to some simplex in the complex, is called an empty simplex. Mikhail Gromov defined the no-Δ condition to be the condition that a complex have no empty simplices. A flag complex is an abstract simplicial complex that has no empty simplices; that is, it is a complex satisfying Gromov's no-Δ condition. Any flag complex is the clique complex of its 1-skeleton. Thus, flag complexes and clique complexes are essentially the same thing. However, in many cases it may be convenient to define a flag complex directly from some data other than a graph, rather than indirectly as the clique complex of a graph derived from that data.
Read more about this topic: Clique Complex
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