Clique complex
![The clique complex of a graph. Cliques of size one are shown as small red disks; cliques of size two are shown as black line segments; cliques of size three are shown as light blue triangles; and cliques of size four are shown as dark blue tetrahedra.](/Images/godic/202501/13/VR_complex.svg0313.png")
Clique complexes, flag complexes, and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph.
The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of G. Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of k vertices is represented by a simplex of dimension k − 1. The 1-skeleton of X(G) (also known as the underlying graph of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2-element set in the family; it is isomorphic to G.