The clique complex 
of a simple graph 
is the abstract
simplicial complex whose nonempty simplices are the
finite cliques of 
. Its 0-simplices are the vertices
of 
, its 1-simplices are the edges
of 
, and its 2-simplices are the triangles
of 
(Kahle 2009).
Every clique complex is a flag complex. Conversely, every flag complex is the clique complex of its 1-skeleton. The terms therefore describe the same class of abstract simplicial complexes. "Clique complex" emphasizes the construction from a graph, while "flag complex" emphasizes the intrinsic property of the resulting complex.
For a finite graph 
,
the dimension of 
is 
, where 
is the clique number
of 
.
If 
is the number of 
-cliques
and 
is the clique polynomial, then the Euler
characteristic is
 |
For example, the clique complex of the complete graph 
is an 
-simplex. The clique complex
of a triangle-free graph contains only vertices
and edges and therefore has dimension at most one.
