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# cliqueComplex -- returns the clique complex of a graph

## Synopsis

• Usage:
D = cliqueComplex G
• Inputs:
• G, ,
• Outputs:
• D, , the clique complex of G

## Description

This function returns the clique complex of a graph $G$. This is the simplicial complex whose faces correspond to the cliques in the graph. That is, $F = \{x_{i_1},...,x_{i_s}\}$ is a face of the clique complex of $G$ if and only if the induced graph on $\{x_{i_1},...,x_{i_s}\}$ is a clique of $G$.

 i1 : R = QQ[w,x,y,z]; i2 : e = graph {w*x,w*y,x*y,y*z} -- clique on {w,x,y} and {y,z} o2 = Graph{"edges" => {{w, x}, {w, y}, {x, y}, {y, z}}} "ring" => R "vertices" => {w, x, y, z} o2 : Graph i3 : cliqueComplex e -- max facets {w,x,y} and {y,z} o3 = simplicialComplex | yz wxy | o3 : SimplicialComplex i4 : g = completeGraph R o4 = Graph{"edges" => {{w, x}, {w, y}, {w, z}, {x, y}, {x, z}, {y, z}}} "ring" => R "vertices" => {w, x, y, z} o4 : Graph i5 : cliqueComplex g o5 = simplicialComplex | wxyz | o5 : SimplicialComplex

## Ways to use cliqueComplex :

• cliqueComplex(Graph)

## For the programmer

The object cliqueComplex is .