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<head><title>edgeIdeal -- creates the edge ideal of a (hyper)graph</title>
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<div><h1>edgeIdeal -- creates the edge ideal of a (hyper)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>I = edgeIdeal H</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>H</tt>, <span>a <a href="___Hyper__Graph.html">hypergraph</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>I</tt>, <span>a <a href="../../Macaulay2Doc/html/___Monomial__Ideal.html">monomial ideal</a></span>, the edge ideal of H</span></li>
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<div class="single"><h2>Description</h2>
<div><p>The edge ideal of a (hyper)graph is a square-free monomial ideal in which the minimal generators correspond to the edges of a (hyper)graph. Along with <a href="_cover__Ideal.html" title="creates the cover ideal of a (hyper)graph">coverIdeal</a>, the function <tt>edgeIdeal</tt> enables us to translate many graph theoretic properties into algebraic properties.</p>
<div>When the input is a finite simple graph, that is, a graph with no loops or multiple edges, then the edge ideal is a quadratic square-free monomial ideal generated by terms of the form <i>x<sub>i</sub>x<sub>j</sub></i> whenever <i>{x<sub>i</sub>,x<sub>j</sub>}</i> is an edge of the graph.</div>
<table class="examples"><tr><td><pre>i1 : S = QQ[a..e];</pre>
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<tr><td><pre>i2 : c5 = cycle S
o2 = Graph{edges => {{a, b}, {b, c}, {c, d}, {d, e}, {a, e}}}
ring => S
vertices => {a, b, c, d, e}
o2 : Graph</pre>
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<tr><td><pre>i3 : edgeIdeal c5
o3 = monomialIdeal (a*b, b*c, c*d, a*e, d*e)
o3 : MonomialIdeal of S</pre>
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<tr><td><pre>i4 : graph flatten entries gens edgeIdeal c5 == c5
o4 = true</pre>
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<tr><td><pre>i5 : k5 = completeGraph S
o5 = Graph{edges => {{a, b}, {a, c}, {a, d}, {a, e}, {b, c}, {b, d}, {b, e}, {c, d}, {c, e}, {d, e}}}
ring => S
vertices => {a, b, c, d, e}
o5 : Graph</pre>
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<tr><td><pre>i6 : edgeIdeal k5
o6 = monomialIdeal (a*b, a*c, b*c, a*d, b*d, c*d, a*e, b*e, c*e, d*e)
o6 : MonomialIdeal of S</pre>
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<div>When the input is a hypergraph, the edge ideal is a square-free monomial ideal generated by monomials of the form <i>x<sub>i<sub>1</sub></sub>x<sub>i<sub>2</sub></sub>...x<sub>i<sub>s</sub></sub></i> whenever <i>{x<sub>i<sub>1</sub></sub>,...,x<sub>i<sub>s</sub></sub>}</i> is an edge of the hypergraph. Because all of our hypergraphs are clutters, that is, no edge is allowed to be a subset of another edge, we have a bijection between the minimal generators of the edge ideal of hypergraph and the edges of the hypergraph.</div>
<table class="examples"><tr><td><pre>i7 : S = QQ[z_1..z_8];</pre>
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<tr><td><pre>i8 : h = hyperGraph {{z_1,z_2,z_3},{z_2,z_3,z_4,z_5},{z_4,z_5,z_6},{z_6,z_7,z_8}}
o8 = HyperGraph{edges => {{z , z , z }, {z , z , z , z }, {z , z , z }, {z , z , z }}}
1 2 3 2 3 4 5 4 5 6 6 7 8
ring => S
vertices => {z , z , z , z , z , z , z , z }
1 2 3 4 5 6 7 8
o8 : HyperGraph</pre>
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<tr><td><pre>i9 : edgeIdeal h
o9 = monomialIdeal (z z z , z z z z , z z z , z z z )
1 2 3 2 3 4 5 4 5 6 6 7 8
o9 : MonomialIdeal of S</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_cover__Ideal.html" title="creates the cover ideal of a (hyper)graph">coverIdeal</a> -- creates the cover ideal of a (hyper)graph</span></li>
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<div class="waystouse"><h2>Ways to use <tt>edgeIdeal</tt> :</h2>
<ul><li>edgeIdeal(HyperGraph)</li>
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