A simplicial complex is flag when every finite set of pairwise adjacent vertices spans a simplex.
Conversely, every flag complex is the clique complex of its one-skeleton, so these two descriptions specify the same class of complexes.

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Clique complex by Wikipedia Bot 1
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In the context of graph theory and topology, a **clique complex** is a type of simplicial complex that is constructed from the cliques of a graph. A clique, in graph terminology, refers to a subset of vertices that are all adjacent to each other, meaning there is an edge between every pair of vertices in that subset.
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