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<head><title>Graph -- a class for graphs</title>
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<div><h1>Graph -- a class for graphs</h1>
<div class="single"><h2>Description</h2>
<div><div>This class represents simple graphs. This class extends <a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a> and hence inherits all HyperGraph methods.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[w,x,y,z];</pre>
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<tr><td><pre>i2 : G = graph(R, {{w,x},{w,y},{w,z},{y,z}});</pre>
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<tr><td><pre>i3 : vertices G
o3 = {w, x, y, z}
o3 : List</pre>
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<tr><td><pre>i4 : edges G
o4 = {{w, x}, {w, y}, {w, z}, {y, z}}
o4 : List</pre>
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<tr><td><pre>i5 : ring G
o5 = R
o5 : PolynomialRing</pre>
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<div>Like hypergraphs, graphs are associated with a polynomial ring whose variables are the vertices of the graph. Isolated vertices should not appear in the edge list. As a consequence, the <a href="_edge__Ideal.html" title="creates the edge ideal of a (hyper)graph">edgeIdeal</a> of a graph is always generated by quadratics. The fact that isolated vertices are not edges in a graph affects the output of the methods <a href="_connected__Components.html" title="returns the connected components of a hypergraph">connectedComponents</a>, <a href="_num__Connected__Components.html" title="returns the number of connected components in a (hyper)graph">numConnectedComponents</a>, and <a href="_is__Connected.html" title="determines if a (hyper)graph is connected">isConnected</a>. One can use <a href="_connected__Graph__Components.html" title="returns the connected components of a graph">connectedGraphComponents</a>, <a href="_num__Connected__Graph__Components.html" title="returns the number of connected components in a graph">numConnectedGraphComponents</a>, and <a href="_is__Connected__Graph.html" title="determines if a graph is connected">isConnectedGraph</a> to ensure that each isolated vertex is counted as a separate connected component.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></li>
<li><span><a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a> -- a class for hypergraphs</span></li>
<li><span><a href="___Constructor_sp__Overview.html" title="a summary of the many ways of making graphs and hypergraphs">Constructor Overview</a> -- a summary of the many ways of making graphs and hypergraphs</span></li>
</ul>
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<div class="waystouse"><h2>Functions and methods returning a graph :</h2>
<ul><li><span><a href="_anti__Cycle.html" title="returns a graph of an anticycle">antiCycle</a> -- returns a graph of an anticycle</span></li>
<li><span>antiCycle(List), see <span><a href="_anti__Cycle.html" title="returns a graph of an anticycle">antiCycle</a> -- returns a graph of an anticycle</span></span></li>
<li><span>antiCycle(Ring), see <span><a href="_anti__Cycle.html" title="returns a graph of an anticycle">antiCycle</a> -- returns a graph of an anticycle</span></span></li>
<li><span>antiCycle(Ring,ZZ), see <span><a href="_anti__Cycle.html" title="returns a graph of an anticycle">antiCycle</a> -- returns a graph of an anticycle</span></span></li>
<li><span>completeGraph(List), see <span><a href="_complete__Graph.html" title="returns a complete graph">completeGraph</a> -- returns a complete graph</span></span></li>
<li><span>completeGraph(Ring), see <span><a href="_complete__Graph.html" title="returns a complete graph">completeGraph</a> -- returns a complete graph</span></span></li>
<li><span>completeGraph(Ring,ZZ), see <span><a href="_complete__Graph.html" title="returns a complete graph">completeGraph</a> -- returns a complete graph</span></span></li>
<li><span>completeMultiPartite(Ring,List), see <span><a href="_complete__Multi__Partite.html" title="returns a complete multipartite graph">completeMultiPartite</a> -- returns a complete multipartite graph</span></span></li>
<li><span>completeMultiPartite(Ring,ZZ,ZZ), see <span><a href="_complete__Multi__Partite.html" title="returns a complete multipartite graph">completeMultiPartite</a> -- returns a complete multipartite graph</span></span></li>
<li><span><a href="_cycle.html" title="returns a graph cycle">cycle</a> -- returns a graph cycle</span></li>
<li><span>cycle(List), see <span><a href="_cycle.html" title="returns a graph cycle">cycle</a> -- returns a graph cycle</span></span></li>
<li><span>cycle(Ring), see <span><a href="_cycle.html" title="returns a graph cycle">cycle</a> -- returns a graph cycle</span></span></li>
<li><span>cycle(Ring,ZZ), see <span><a href="_cycle.html" title="returns a graph cycle">cycle</a> -- returns a graph cycle</span></span></li>
<li><span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></li>
<li><span>graph(HyperGraph), see <span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></span></li>
<li><span>graph(Ideal), see <span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></span></li>
<li><span>graph(List), see <span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></span></li>
<li><span>graph(MonomialIdeal), see <span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></span></li>
<li><span>graph(PolynomialRing,List), see <span><a href="_graph.html" title="constructor for Graph">graph</a> -- constructor for Graph</span></span></li>
</ul>
<h2>Methods that use a graph :</h2>
<ul><li><span>adjacencyMatrix(Graph), see <span><a href="_adjacency__Matrix.html" title="returns the adjacency Matrix of a graph">adjacencyMatrix</a> -- returns the adjacency Matrix of a graph</span></span></li>
<li><span>allEvenHoles(Graph), see <span><a href="_all__Even__Holes.html" title="returns all even holes in a graph">allEvenHoles</a> -- returns all even holes in a graph</span></span></li>
<li><span>allOddHoles(Graph), see <span><a href="_all__Odd__Holes.html" title="returns all odd holes in a graph">allOddHoles</a> -- returns all odd holes in a graph</span></span></li>
<li><span>cliqueComplex(Graph), see <span><a href="_clique__Complex.html" title="returns the clique complex of a graph">cliqueComplex</a> -- returns the clique complex of a graph</span></span></li>
<li><span>cliqueNumber(Graph), see <span><a href="_clique__Number.html" title="computes the clique number of a graph">cliqueNumber</a> -- computes the clique number of a graph</span></span></li>
<li><span>complementGraph(Graph), see <span><a href="_complement__Graph.html" title="returns the complement of a graph or hypergraph">complementGraph</a> -- returns the complement of a graph or hypergraph</span></span></li>
<li><span>getCliques(Graph), see <span><a href="_get__Cliques.html" title="returns cliques in a graph">getCliques</a> -- returns cliques in a graph</span></span></li>
<li><span>getCliques(Graph,ZZ), see <span><a href="_get__Cliques.html" title="returns cliques in a graph">getCliques</a> -- returns cliques in a graph</span></span></li>
<li><span>getMaxCliques(Graph), see <span><a href="_get__Max__Cliques.html" title="returns maximal cliques in a graph">getMaxCliques</a> -- returns maximal cliques in a graph</span></span></li>
<li><span>hasOddHole(Graph), see <span><a href="_has__Odd__Hole.html" title="tells whether a graph contains an odd hole">hasOddHole</a> -- tells whether a graph contains an odd hole</span></span></li>
<li><span>hyperGraph(Graph), see <span><a href="_hyper__Graph.html" title="constructor for HyperGraph">hyperGraph</a> -- constructor for HyperGraph</span></span></li>
<li><span>independenceNumber(Graph), see <span><a href="_independence__Number.html" title="determines the independence number of a graph">independenceNumber</a> -- determines the independence number of a graph</span></span></li>
<li><span>inducedGraph(Graph,List), see <span><a href="_induced__Graph.html" title="returns the induced subgraph of a graph">inducedGraph</a> -- returns the induced subgraph of a graph</span></span></li>
<li><span>isBipartite(Graph), see <span><a href="_is__Bipartite.html" title="determines if a graph is bipartite">isBipartite</a> -- determines if a graph is bipartite</span></span></li>
<li><span>isChordal(Graph), see <span><a href="_is__Chordal.html" title="determines if a graph is chordal">isChordal</a> -- determines if a graph is chordal</span></span></li>
<li><span>isForest(Graph), see <span><a href="_is__Forest.html" title="determines whether a (hyper)graph is a forest">isForest</a> -- determines whether a (hyper)graph is a forest</span></span></li>
<li><span>isLeaf(Graph,ZZ), see <span><a href="_is__Leaf.html" title="determines if an edge (or vertex) is a leaf of a (hyper)graph">isLeaf</a> -- determines if an edge (or vertex) is a leaf of a (hyper)graph</span></span></li>
<li><span>isPerfect(Graph), see <span><a href="_is__Perfect.html" title="determines whether a graph is perfect">isPerfect</a> -- determines whether a graph is perfect</span></span></li>
<li><span>numTriangles(Graph), see <span><a href="_num__Triangles.html" title="returns the number of triangles in a graph">numTriangles</a> -- returns the number of triangles in a graph</span></span></li>
<li><span>smallestCycleSize(Graph), see <span><a href="_smallest__Cycle__Size.html" title="returns the size of the smallest induced cycle of a graph">smallestCycleSize</a> -- returns the size of the smallest induced cycle of a graph</span></span></li>
<li><span>spanningTree(Graph), see <span><a href="_spanning__Tree.html" title="returns a spanning tree of a graph">spanningTree</a> -- returns a spanning tree of a graph</span></span></li>
</ul>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Graph.html" title="a class for graphs">Graph</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="___Hyper__Graph.html" title="a class for hypergraphs">HyperGraph</a> < <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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