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<head><title>smallestCycleSize -- returns the size of the smallest induced cycle of a graph</title>
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<div><h1>smallestCycleSize -- returns the size of the smallest induced cycle 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>s = smallestCycleSize(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>, the input</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>s</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the size of the smallest induced cycle</span></li>
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<div class="single"><h2>Description</h2>
<div><div>This function returns the size of the smallest induced cycle of a graph. It is based upon Theorem 2.1 in the paper "Restricting linear syzygies: algebra and geometry" by Eisenbud, Green, Hulek, and Popsecu. This theorem states that if G is graph, then the edge ideal of the complement of G satisfies property N<sub>2,p</sub>, that is, the resolution of I(G<sup>c</sup>) is linear up to the p-th step, if and only if the smallest induced cycle of G has length p+3. The algorithm looks at the resolution of the edge ideal of the complement to determine the size of the smallest cycle.</div>
<table class="examples"><tr><td><pre>i1 : T = QQ[x_1..x_9];</pre>
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<tr><td><pre>i2 : g = graph {x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_5*x_6,x_6*x_7,x_7*x_8,x_8*x_9,x_9*x_1} -- a 9-cycle
o2 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 3 4 4 5 5 6 6 7 7 8 1 9 8 9
ring => T
vertices => {x , x , x , x , x , x , x , x , x }
1 2 3 4 5 6 7 8 9
o2 : Graph</pre>
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<tr><td><pre>i3 : smallestCycleSize g
o3 = 9</pre>
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<tr><td><pre>i4 : h = graph {x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_5*x_6,x_6*x_7,x_7*x_8,x_8*x_9} -- a tree (no cycles)
o4 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9
ring => T
vertices => {x , x , x , x , x , x , x , x , x }
1 2 3 4 5 6 7 8 9
o4 : Graph</pre>
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<tr><td><pre>i5 : smallestCycleSize h
o5 = infinity
o5 : InfiniteNumber</pre>
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<tr><td><pre>i6 : l = graph {x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_5,x_5*x_6,x_6*x_7,x_7*x_8,x_8*x_9,x_9*x_1,x_1*x_4}
o6 = Graph{edges => {{x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }, {x , x }}}
1 2 2 3 1 4 3 4 4 5 5 6 6 7 7 8 1 9 8 9
ring => T
vertices => {x , x , x , x , x , x , x , x , x }
1 2 3 4 5 6 7 8 9
o6 : Graph</pre>
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<tr><td><pre>i7 : smallestCycleSize l
o7 = 4</pre>
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<div>Note that <tt>G</tt> is a tree if and only if <tt>smallestCycleSize G</tt> is <tt>infinity</tt>.</div>
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<div class="waystouse"><h2>Ways to use <tt>smallestCycleSize</tt> :</h2>
<ul><li>smallestCycleSize(Graph)</li>
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