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<head><title>cliqueComplex -- returns the clique complex of a graph</title>
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<div><h1>cliqueComplex -- returns the clique complex of a graph</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>D = cliqueComplex G</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>a <a href="___Graph.html">graph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>D</tt>, <span>a <a href="../../SimplicialComplexes/html/___Simplicial__Complex.html">simplicial complex</a></span>, the clique complex of <tt>G</tt></span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function returns the clique complex of a graph <i>G</i>. This is the simplicial complex whose faces correspond to the cliques in the graph. That is, <i>F = {x<sub>i<sub>1</sub></sub>,...,x<sub>i<sub>s</sub></sub>}</i> is a face of the clique complex of <i>G</i> if and only if the induced graph on <i>{x<sub>i<sub>1</sub></sub>,...,x<sub>i<sub>s</sub></sub>}</i> is a clique of <i>G</i>.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[w,x,y,z];</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i3 : cliqueComplex e -- max facets {w,x,y} and {y,z}
o3 = | yz wxy |
o3 : SimplicialComplex</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i5 : cliqueComplex g
o5 = | wxyz |
o5 : SimplicialComplex</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_clique__Number.html" title="computes the clique number of a graph">cliqueNumber</a> -- computes the clique number of a graph</span></li>
<li><span><a href="_get__Cliques.html" title="returns cliques in a graph">getCliques</a> -- returns cliques in a graph</span></li>
<li><span><a href="_get__Max__Cliques.html" title="returns maximal cliques in a graph">getMaxCliques</a> -- returns maximal cliques in a graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>cliqueComplex</tt> :</h2>
<ul><li>cliqueComplex(Graph)</li>
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