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<!doctype html> <html lang="en" class="discretemathematics"> <head> <title>Clique -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Clique" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="A clique of a graph G is a complete subgraph of G, and the clique of largest possible size is referred to as a maximum clique (which has size known as the (upper) clique number omega(G)). However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not..." /> <meta name="description" content="A clique of a graph G is a complete subgraph of G, and the clique of largest possible size is referred to as a maximum clique (which has size known as the (upper) clique number omega(G)). However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2010-09-29" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2010-10-25" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2011-01-11" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2011-12-09" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2012-12-12" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2013-06-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-01-06" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2019-06-10" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Cliques" /> <meta name="DC.Subject" scheme="MSC_2000" content="05C69" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. 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However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Clique -- from Wolfram MathWorld"> <meta name="twitter:description" content="A clique of a graph G is a complete subgraph of G, and the clique of largest possible size is referred to as a maximum clique (which has size known as the (upper) clique number omega(G)). However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A maximal clique is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_Clique.png"> <link rel="canonical" href="https://mathworld.wolfram.com/Clique.html" /> <meta http-equiv="x-ua-compatible" content="ie=edge"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta charset="utf-8"> <script async src="/common/javascript/analytics.js"></script> <script async src="//www.wolframcdn.com/consent/cookie-consent.js"></script> <script async src="/common/javascript/wal/latest/walLoad.js"></script> <link rel="stylesheet" href="/css/styles.css"> <link rel="preload" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css" as="style" onload="this.onload=null;this.rel='stylesheet'"> <noscript><link rel="stylesheet" href="//www.wolframcdn.com/fonts/source-sans-pro/1.0/global.css"></noscript> </head> <body id="topics"> <main id="entry"> <div class="wrapper"> <section id="container"> 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method="get" action="/search/" name="search" id="search-mobile" accept-charset="UTF-8"> <input type="text" name="query" placeholder="Search" id="searchFieldMobile"> <img src="/images/header/search-icon.png" width="18" height="18" id="mobile-search" class="search-btn" title="Search" alt="Search" onclick="submitForm()"> </form> <nav class="topics-nav"> <a href="/topics/Algebra.html" id="sidebar-algebra"> Algebra </a> <a href="/topics/AppliedMathematics.html" id="sidebar-appliedmathematics"> Applied Mathematics </a> <a href="/topics/CalculusandAnalysis.html" id="sidebar-calculusandanalysis"> Calculus and Analysis </a> <a href="/topics/DiscreteMathematics.html" id="sidebar-discretemathematics"> Discrete Mathematics </a> <a href="/topics/FoundationsofMathematics.html" id="sidebar-foundationsofmathematics"> Foundations of Mathematics </a> <a href="/topics/Geometry.html" id="sidebar-geometry"> Geometry </a> <a href="/topics/HistoryandTerminology.html" id="sidebar-historyandterminology"> History and Terminology </a> <a href="/topics/NumberTheory.html" id="sidebar-numbertheory"> Number Theory </a> <a href="/topics/ProbabilityandStatistics.html" id="sidebar-probabilityandstatistics"> Probability and Statistics </a> <a href="/topics/RecreationalMathematics.html" id="sidebar-recreationalmathematics"> Recreational Mathematics </a> <a href="/topics/Topology.html" id="sidebar-topology"> Topology </a> </nav> <nav class="secondary-nav"> <a href="/letters/"> Alphabetical Index </a> <a href="/whatsnew/"> New in MathWorld </a> </nav> </section> <section id="content">  <nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/Cliques.html">Cliques</a> </li> </ul></nav>   <h1>Clique</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/GraphTheory/Clique.nb" download="Clique.nb"><img src="/images/entries/download-notebook-icon.png" width="26" height="27" alt="DOWNLOAD Mathematica Notebook" /><span>Download <span class="display-i display-n__600">Wolfram </span>Notebook</span></a> </div>  <div class="entry-content"> <div class="center-image"> <img style="max-width:100%;max-height:100%;" width="502.905" height="97.9039" src="images/eps-svg/Clique_950.svg" class="" alt="Clique" /> </div> <p> A clique of a graph <img src="/images/equations/Clique/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is a complete subgraph of <img src="/images/equations/Clique/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, and the clique of largest possible size is referred to as a <a href="/MaximumClique.html">maximum clique</a> (which has size known as the (upper) <a href="/CliqueNumber.html">clique number</a> <img src="/images/equations/Clique/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="omega(G)" />). However, care is needed since maximum cliques are often called simply "cliques" (e.g., Harary 1994). A <a href="/MaximalClique.html">maximal clique</a> is a clique that cannot be extended by including one more adjacent vertex, meaning it is not a subset of a larger clique. Maximum cliques are therefore maximal cliqued (but not necessarily vice versa). </p> <p> Cliques arise in a number of areas of <a href="/GraphTheory.html">graph theory</a> and combinatorics, including graph coloring and the theory of <a href="/Error-CorrectingCode.html">error-correcting codes</a>. </p> <p> A clique of size <img src="/images/equations/Clique/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> is called a <img src="/images/equations/Clique/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />-clique (though this term is also sometimes used to mean a maximal set of vertices that are at a distance no greater than <img src="/images/equations/Clique/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> from each other). 0-cliques correspond to the empty set (sets of 0 vertices), 1-cliques correspond to vertices, 2-cliques to edges, and 3-cliques to 3-cycles. </p> <p> The <a href="/CliquePolynomial.html">clique polynomial</a> is of a graph <img src="/images/equations/Clique/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is defined as </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/Clique/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="130" height="54" alt=" C_G(x)=sum_(k=0)^(omega(G))c_kx^k, " /></td></tr> </table> </div> <p> where <img src="/images/equations/Clique/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="15" height="21" alt="c_k" /> is the number of cliques of size <img src="/images/equations/Clique/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" />, with <img src="/images/equations/Clique/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="22" alt="c_0=1" />, <img src="/images/equations/Clique/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="57" height="21" alt="c_1=|G|" /> equal to the <a href="/VertexCount.html">vertex count</a> of <img src="/images/equations/Clique/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, <img src="/images/equations/Clique/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="76" height="21" alt="c_2=m(G)" /> equal to the <a href="/EdgeCount.html">edge count</a> of <img src="/images/equations/Clique/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, etc. </p> <p> In the <a href="http://www.wolfram.com/language/">Wolfram Language</a>, the command <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>][[1]] can be used to find a <a href="/MaximumClique.html">maximum clique</a>, and <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <tt>Length</tt> /@ <tt>FindClique</tt>[g], <tt>All</tt>] to find all <a href="/MaximumClique.html">maximum cliques</a>. Similarly, <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <tt>Infinity</tt>] can be used to find a <a href="/MaximalClique.html">maximal clique</a>, and <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <tt>Infinity</tt>, <tt>All</tt>] to find all maximal cliques. To find all cliques, enumerate all vertex subsets <img src="/images/equations/Clique/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="7" height="21" alt="s" /> and select those for which <tt><a href="http://reference.wolfram.com/language/ref/CompleteGraphQ.html">CompleteGraphQ</a></tt>[<i>g</i>, <i>s</i>] is true. </p> <p> In general, <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <i>n</i>] can be used to find a <a href="/MaximalClique.html">maximal clique</a> containing at least <img src="/images/equations/Clique/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> vertices, <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <i>n</i>, <tt>All</tt>] to find all such cliques, <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <img src="/images/equations/Clique/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="{" /><i>n</i><img src="/images/equations/Clique/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="}" />] to find a clique containing at exactly <img src="/images/equations/Clique/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> vertices, and <tt><a href="http://reference.wolfram.com/language/ref/FindClique.html">FindClique</a></tt>[<i>g</i>, <img src="/images/equations/Clique/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="{" /><i>n</i><img src="/images/equations/Clique/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="}" />, <tt>All</tt>] to find all such cliques. </p> <p> The numbers of cliques, equal to the <a href="/CliquePolynomial.html">clique polynomial</a> evaluated at <img src="/images/equations/Clique/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="x=1" />, for various members of graph families are summarized in the table below, where the trivial 0-clique represented by the initial 1 in the <a href="/CliquePolynomial.html">clique polynomial</a> is included in each count. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph family</td><td align="left">OEIS</td><td align="left">number of cliques</td></tr><tr style=""><td align="left"><a href="/AlternatingGroupGraph.html">alternating group graph</a></td><td align="left"><a href="http://oeis.org/A308599">A308599</a></td><td align="left">X, 2, 8, 45, 301, 2281, ...</td></tr><tr style=""><td align="left"><a href="/AndrasfaiGraph.html">Andrásfai graph</a></td><td align="left"><a href="http://oeis.org/A115067">A115067</a></td><td align="left">4, 11, 21, 34, 50, 69, 91, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/AntelopeGraph.html">antelope graph</a></td><td align="left"><a href="http://oeis.org/A308600">A308600</a></td><td align="left">2, 5, 10, 17, 34, 61, 98, ...</td></tr><tr style=""><td align="left"><a href="/AntiprismGraph.html">antiprism graph</a></td><td align="left"><a href="http://oeis.org/A017077">A017077</a></td><td align="left">X, X, 27, 33, 41, 49, 57, 65, ...</td></tr><tr style=""><td align="left"><a href="/ApollonianNetwork.html">Apollonian network</a></td><td align="left"><a href="http://oeis.org/A205248">A205248</a></td><td align="left">16, 40, 112, 328, 976, 2920, ...</td></tr><tr style=""><td align="left"><a href="/BarbellGraph.html">barbell graph</a></td><td align="left"><a href="http://oeis.org/A000079">A000079</a></td><td align="left">X, X, 16, 32, 64, 128, 256, 512, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/BishopGraph.html">bishop graph</a></td><td align="left"><a href="http://oeis.org/A183156">A183156</a></td><td align="left">2, 7, 22, 59, 142, 319, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/BlackBishopGraph.html">black bishop graph</a></td><td align="left"><a href="http://oeis.org/A295909">A295909</a></td><td align="left">2, 4, 14, 30, 82, 160, 386, ...</td></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/Clique/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="22" alt="S_(n+1) square P_2" /></td><td align="left"><a href="http://oeis.org/A016897">A016897</a></td><td align="left">9, 14, 19, 24, 29, 34, 39, 44, ...</td></tr><tr style=""><td align="left"><a href="/BruhatGraph.html">Bruhat graph</a></td><td align="left"><a href="http://oeis.org/A139149">A139149</a></td><td align="left">2, 4, 13, 61, 361, 2521, 20161, ...</td></tr><tr style=""><td align="left"><a href="/CentipedeGraph.html">centipede graph</a></td><td align="left"><a href="http://oeis.org/A008586">A008586</a></td><td align="left">4, 8, 12, 16, 20, 24, 28, 32, 36, ...</td></tr><tr style=""><td align="left"><a href="/CocktailPartyGraph.html">cocktail party graph</a> <img src="/images/equations/Clique/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="22" alt="K_(n×2)" /></td><td align="left"><a href="http://oeis.org/A000244">A000244</a></td><td align="left">3, 9, 27, 81, 243, 729, 2187, ...</td></tr><tr style=""><td align="left"><a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/Clique/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /></td><td align="left"><a href="http://oeis.org/A000079">A000079</a></td><td align="left">2, 4, 8, 16, 32, 64, 128, 256, ...</td></tr><tr style=""><td align="left"><a href="/CompleteBipartiteGraph.html">complete bipartite graph</a> <img src="/images/equations/Clique/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(n,n)" /></td><td align="left"><a href="http://oeis.org/A000290">A000290</a></td><td align="left">4, 9, 16, 25, 36, 49, 64, 81, 100, ...</td></tr><tr style=""><td align="left"><a href="/CompleteTripartiteGraph.html">complete tripartite graph</a> <img src="/images/equations/Clique/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="23" alt="K_(n,n,n)" /></td><td align="left"><a href="http://oeis.org/A000578">A000578</a></td><td align="left">8, 27, 64, 125, 216, 343, 512, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="2n" />-<a href="/CrossedPrismGraph.html">crossed prism graph</a></td><td align="left"><a href="http://oeis.org/A017281">A017281</a></td><td align="left">X, 21, 31, 41, 51, 61, 71, ...</td></tr><tr style=""><td align="left"><a href="/CrownGraph.html">crown graph</a> <img src="/images/equations/Clique/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="K_2 square K_n^_" /></td><td align="left"><a href="http://oeis.org/A002061">A002061</a></td><td align="left">X, X, 13, 21, 31, 43, 57, 73, 91, ...</td></tr><tr style=""><td align="left"><a href="/Cube-ConnectedCycleGraph.html">cube-connected cycle graph</a></td><td align="left"><a href="http://oeis.org/A295926">A295926</a></td><td align="left">X, X, 69, 161, 401, 961, 2241, 5121, ...</td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/Clique/Inline33.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="left"><a href="http://oeis.org/A308602">A308602</a></td><td align="left">X, X, 8, 9, 11, 13, 15, 17, 19, ...</td></tr><tr style=""><td align="left"><a href="/DipyramidalGraph.html">dipyramidal graph</a></td><td align="left"><a href="http://oeis.org/A308603">A308603</a></td><td align="left">X, X, 24, 27, 33, 39, 45, 51, 57, 63, ...</td></tr><tr style=""><td align="left"><a href="/EmptyGraph.html">empty graph</a> <img src="/images/equations/Clique/Inline34.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="K^__n" /></td><td align="left"><a href="http://oeis.org/A000027">A000027</a></td><td align="left">2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...</td></tr><tr style=""><td align="left"><a href="/FibonacciCubeGraph.html">Fibonacci cube graph</a></td><td align="left"><a href="http://oeis.org/A291916">A291916</a></td><td align="left">4, 6, 11, 19, 34, 60, 106, 186, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline35.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/FiveleaperGraph.html">fiveleaper graph</a></td><td align="left"><a href="http://oeis.org/A308604">A308604</a></td><td align="left">X, 4, 16, 25, 57, 129, 289, 641, 1409, ...</td></tr><tr style=""><td align="left"><a href="/FoldedCubeGraph.html">folded cube graph</a></td><td align="left"><a href="http://oeis.org/A295921">A295921</a></td><td align="left">3, 15, 24, 56, ...</td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="left"><a href="http://oeis.org/A016873">A016873</a></td><td align="left">X, X, 17, 22, 27, 32, 37, 42, 47, 52, ...</td></tr><tr style=""><td align="left"><a href="/GridGraph.html">grid graph</a> <img src="/images/equations/Clique/Inline36.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_n square P_n" /></td><td align="left"><a href="http://oeis.org/A056105">A056105</a></td><td align="left">2, 9, 22, 41, 66, 97, 134, 177, 226, 281, ...</td></tr><tr style=""><td align="left"><a href="/GridGraph.html">grid graph</a> <img src="/images/equations/Clique/Inline37.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="89" height="21" alt="P_n square P_n square P_n" /></td><td align="left"><a href="http://oeis.org/A295907">A295907</a></td><td align="left">2, 21, 82, 209, 426, 757, 1226, 1857, ...</td></tr><tr style=""><td align="left"><a href="/HalvedCubeGraph.html">halved cube graph</a></td><td align="left"><a href="http://oeis.org/A295922">A295922</a></td><td align="left">2, 4, 16, 81, 393, 1777, ...</td></tr><tr style=""><td align="left"><a href="/HanoiGraph.html">Hanoi graph</a></td><td align="left"><a href="http://oeis.org/A295911">A295911</a></td><td align="left">8, 25, 76, 229, 688, ...</td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="left"><a href="http://oeis.org/A016933">A016933</a></td><td align="left">X, X, 22, 26, 32, 38, 44, 50, 56, ...</td></tr><tr style=""><td align="left"><a href="/HypercubeGraph.html">hypercube graph</a> <img src="/images/equations/Clique/Inline38.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="Q_n" /></td><td align="left"><a href="http://oeis.org/A132750">A132750</a></td><td align="left">4, 9, 21, 49, 113, 257, 577, 1281, 2817, ...</td></tr><tr style=""><td align="left"><a href="/KellerGraph.html">Keller graph</a></td><td align="left"><a href="http://oeis.org/A295902">A295902</a></td><td align="left">5, 57, 14833, 2290312801, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline39.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/KingGraph.html">king graph</a></td><td align="left"><a href="http://oeis.org/A295906">A295906</a></td><td align="left">2, 16, 50, 104, 178, 272, 386, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline40.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/KnightGraph.html">knight graph</a></td><td align="left"><a href="http://oeis.org/A295905">A295905</a></td><td align="left">2, 5, 18, 41, 74, 117, 170, 233, 306, 389, ...</td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a> <img src="/images/equations/Clique/Inline41.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="P_2 square P_n" /></td><td align="left"><a href="http://oeis.org/A016897">A016897</a></td><td align="left">4, 9, 14, 19, 24, 29, 34, 39, 44, 49, 54, ...</td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a> <img src="/images/equations/Clique/Inline42.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="nP_2" /></td><td align="left"><a href="http://oeis.org/A016777">A016777</a></td><td align="left">4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, ...</td></tr><tr style=""><td align="left"><a href="/MengerSpongeGraph.html">Menger sponge graph</a></td><td align="left"><a href="http://oeis.org/A292209">A292209</a></td><td align="left">45, 1073, 22977, ...</td></tr><tr style=""><td align="left"><a href="/MoebiusLadder.html">Möbius ladder</a> <img src="/images/equations/Clique/Inline43.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="M_n" /></td><td align="left"><a href="http://oeis.org/A016861">A016861</a></td><td align="left">X, X, 16, 21, 26, 31, 36, 41, 46, 51, ...</td></tr><tr style=""><td align="left"><a href="/MycielskiGraph.html">Mycielski graph</a></td><td align="left"><a href="http://oeis.org/A199109">A199109</a></td><td align="left">2, 4, 11, 32, 95, 284, 851, 2552, 7655, ...</td></tr><tr style=""><td align="left"><a href="/OddGraph.html">odd graph</a> <img src="/images/equations/Clique/Inline44.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="O_n" /></td><td align="left"><a href="http://oeis.org/A295934">A295934</a></td><td align="left">2, 8, 26, 106, 442, 1849, 7723, ...</td></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="left"><a href="http://oeis.org/A005408">A005408</a></td><td align="left">X, X, 10, 11, 13, 15, 17, 19, 21, 23, ...</td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/Clique/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_n" /></td><td align="left"><a href="http://oeis.org/A005843">A005843</a></td><td align="left">2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...</td></tr><tr style=""><td align="left"><a href="/PathComplementGraph.html">path complement graph</a> <img src="/images/equations/Clique/Inline46.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="P^__n" /></td><td align="left"><a href="http://oeis.org/A000045">A000045</a></td><td align="left">2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...</td></tr><tr style=""><td align="left"><a href="/PermutationStarGraph.html">permutation star graph</a></td><td align="left"><a href="http://oeis.org/A139149">A139149</a></td><td align="left">2, 4, 13, 61, 361, 2521, ...</td></tr><tr style=""><td align="left"><a href="/PolygonDiagonalIntersectionGraph.html">polygon diagonal intersection graph</a></td><td align="left"><a href="http://oeis.org/A300524">A300524</a></td><td align="left">X, X, 8, 18, 41, 80, 169, 250, ...</td></tr><tr style=""><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/Clique/Inline47.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="55" height="21" alt="K_2 square C_n" /></td><td align="left"><a href="http://oeis.org/A016861">A016861</a></td><td align="left">X, X, 18, 21, 26, 31, 36, 41, 46, 51, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline48.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/QueenGraph.html">queen graph</a></td><td align="left"><a href="http://oeis.org/A295903">A295903</a></td><td align="left">2, 16, 94, 293, 742, 1642, 3458, 7087, ...</td></tr><tr style=""><td align="left"><a href="/RookGraph.html">rook graph</a> <img src="/images/equations/Clique/Inline49.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="K_n square K_n" /></td><td align="left"><a href="http://oeis.org/A288958">A288958</a></td><td align="left">2, 9, 34, 105, 286, 721, 1730, ...</td></tr><tr style=""><td align="left"><a href="/RookComplementGraph.html">rook complement graph</a> <img src="/images/equations/Clique/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="56" height="21" alt="K_n square K_n^_" /></td><td align="left"><a href="http://oeis.org/A002720">A002720</a></td><td align="left">2, 7, 34, 209, 1546, 13327, 130922, ...</td></tr><tr style=""><td align="left"><a href="/SierpinskiCarpetGraph.html">Sierpiński carpet graph</a></td><td align="left"><a href="http://oeis.org/A295932">A295932</a></td><td align="left">17, 153, 1289, 10521, ...</td></tr><tr style=""><td align="left"><a href="/SierpinskiGasketGraph.html">Sierpiński gasket graph</a></td><td align="left"><a href="http://oeis.org/A295933">A295933</a></td><td align="left">8, 20, 55, 160, 475, ...</td></tr><tr style=""><td align="left"><a href="/SierpinskiTetrahedronGraph.html">Sierpiński tetrahedron graph</a></td><td align="left"><a href="http://oeis.org/A292537">A292537</a></td><td align="left">6, 59, 227, 899, 3587, 14339, ...</td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/Clique/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="left"><a href="http://oeis.org/A005843">A005843</a></td><td align="left">2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ...</td></tr><tr style=""><td align="left"><a href="/SunGraph.html">sun graph</a></td><td align="left"><a href="http://oeis.org/A295904">A295904</a></td><td align="left">X, X, 20, 32, 52, 88, 156, 288, 548, ...</td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/Clique/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="C_n circledot K_1" /></td><td align="left"><a href="http://oeis.org/A016813">A016813</a></td><td align="left">X, X, 14, 17, 21, 25, 29, 33, 37, 41, 45, ...</td></tr><tr style=""><td align="left"><a href="/TetrahedralGraph.html">tetrahedral graph</a></td><td align="left"><a href="http://oeis.org/A289837">A289837</a></td><td align="left">X, X, X, X, X, 261, 757, 2003, 5035, ...</td></tr><tr style=""><td align="left"><a href="/TorusGridGraph.html">torus grid graph</a> <img src="/images/equations/Clique/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="C_n square C_n" /></td><td align="left"><a href="http://oeis.org/A056107">A056107</a></td><td align="left">X, X, 34, 49, 76, 109, 148, 193, ...</td></tr><tr style=""><td align="left"><a href="/TranspositionGraph.html">transposition graph</a></td><td align="left"><a href="http://oeis.org/A308606">A308606</a></td><td align="left">2, 4, 16, 97, 721, 6121, ...</td></tr><tr style=""><td align="left"><a href="/TriangularGraph.html">triangular graph</a></td><td align="left"><a href="http://oeis.org/A290056">A290056</a></td><td align="left">X, 2, 8, 27, 76, 192, 456, 1045, ...</td></tr><tr style=""><td align="left"><a href="/TriangularGridGraph.html">triangular grid graph</a></td><td align="left"><a href="http://oeis.org/A242658">A242658</a></td><td align="left">8, 20, 38, 62, 92, 128, 170, 218, ...</td></tr><tr style=""><td align="left"><a href="/TriangularSnakeGraph.html">triangular snake graph</a> <img src="/images/equations/Clique/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="TS_n" /></td><td align="left"><a href="http://oeis.org/A016789">A016789</a></td><td align="left">2, X, 8, X, 14, X, 20, X, 26, X, 32, X, ...</td></tr><tr style=""><td align="left"><a href="/WebGraph.html">web graph</a></td><td align="left"><a href="http://oeis.org/A016993">A016993</a></td><td align="left">X, X, 24, 29, 36, 43, 50, 57, 64, 71, 78, ...</td></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/Clique/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="W_n" /></td><td align="left"><a href="http://oeis.org/A308607">A308607</a></td><td align="left">X, X, X, 16, 18, 22, 26, 30, 34, 38, 42, 46, ...</td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline56.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt="n×n" /> <a href="/WhiteBishopGraph.html">white bishop graph</a></td><td align="left"><a href="http://oeis.org/A295910">A295910</a></td><td align="left">X, 4, 9, 30, 61, 160, 301, 71, ...</td></tr> </table> </div> <p> Closed forms for some of these are summarized in the table below. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph family</td><td align="left">number of cliques</td></tr><tr style=""><td align="left"><a href="/AndrasfaiGraph.html">Andrásfai graph</a></td><td align="left"><img src="/images/equations/Clique/Inline57.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="127" height="27" alt="1/2(n+2)(3n-1)" /></td></tr><tr style=""><td align="left"><a href="/AntiprismGraph.html">antiprism graph</a></td><td align="left"><img src="/images/equations/Clique/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{26 for n=3; 8n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/Clique/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="65" height="22" alt="S_(n+1) square P_2" /></td><td align="left"><img src="/images/equations/Clique/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="5n+3" /></td></tr><tr style=""><td align="left"><a href="/CocktailPartyGraph.html">cocktail party graph</a> <img src="/images/equations/Clique/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="34" height="22" alt="K_(n×2)" /></td><td align="left"><img src="/images/equations/Clique/Inline62.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="3^n-1" /></td></tr><tr style=""><td align="left"><a href="/CompleteBipartiteGraph.html">complete bipartite graph</a> <img src="/images/equations/Clique/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(n,n)" /></td><td align="left"><img src="/images/equations/Clique/Inline64.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="44" height="21" alt="2^n-1" /></td></tr><tr style=""><td align="left"><a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/Clique/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /></td><td align="left"><img src="/images/equations/Clique/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="21" alt="n(n+2)" /></td></tr><tr style=""><td align="left"><a href="/CompleteTripartiteGraph.html">complete tripartite graph</a> <img src="/images/equations/Clique/Inline67.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="23" alt="K_(n,n,n)" /></td><td align="left"><img src="/images/equations/Clique/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="108" height="25" alt="n(n^2+3n+3)" /></td></tr><tr style=""><td align="left"><img src="/images/equations/Clique/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="2n" />-<a href="/CrossedPrismGraph.html">crossed prism graph</a></td><td align="left"><img src="/images/equations/Clique/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="31" height="21" alt="10n" /></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/Clique/Inline71.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="left"><img src="/images/equations/Clique/Inline72.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{7 for n=3; 2n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/EmptyGraph.html">empty graph</a> <img src="/images/equations/Clique/Inline73.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="K^__n" /></td><td align="left"><img src="/images/equations/Clique/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /></td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="left"><img src="/images/equations/Clique/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="5n+11" /></td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="left"><img src="/images/equations/Clique/Inline76.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{21 for n=3; 8n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/HypercubeGraph.html">hypercube graph</a> <img src="/images/equations/Clique/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="22" alt="Q_n" /></td><td align="left"><img src="/images/equations/Clique/Inline78.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="81" height="21" alt="2^(n-1)(n+2)" /></td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a></td><td align="left"><img src="/images/equations/Clique/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="5n-2" /></td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a> <img src="/images/equations/Clique/Inline80.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="nP_2" /></td><td align="left"><img src="/images/equations/Clique/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="3n" /></td></tr><tr style=""><td align="left"><a href="/MoebiusLadder.html">Möbius ladder</a> <img src="/images/equations/Clique/Inline82.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="M_n" /></td><td align="left"><img src="/images/equations/Clique/Inline83.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="21" alt="5(n+2)" /></td></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="left"><img src="/images/equations/Clique/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="192" height="61" alt="{9 for n=3; 2(n+1) for n>=4" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/Clique/Inline85.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_n" /></td><td align="left"><img src="/images/equations/Clique/Inline86.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="2n-1" /></td></tr><tr style=""><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/Clique/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="Y_n" /></td><td align="left"><img src="/images/equations/Clique/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{17 for n=3; 5n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/Clique/Inline89.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="left"><img src="/images/equations/Clique/Inline90.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="50" height="21" alt="2n-1" /></td></tr><tr style=""><td align="left"><a href="/SunGraph.html">sun graph</a></td><td align="left"><img src="/images/equations/Clique/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="85" height="21" alt="2^n+4n-1" /></td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/Clique/Inline92.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="59" height="21" alt="C_n circledot K_1" /></td><td align="left"><img src="/images/equations/Clique/Inline93.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{13 for n=3; 4n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/WebGraph.html">web graph</a></td><td align="left"><img src="/images/equations/Clique/Inline94.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="134" height="61" alt="{23 for n=3; 7n for n>=4" /></td></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/Clique/Inline95.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="W_n" /></td><td align="left"><img src="/images/equations/Clique/Inline96.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="168" height="61" alt="{15 for n=3; 4n-3 for n>=4" /></td></tr> </table> </div> </div>  <hr class="margin-b-1-1-4"> <div class="c-777 entry-secondary-content">  <h2>See also</h2><a href="/CliqueCovering.html">Clique Covering</a>, <a href="/CliqueCoveringNumber.html">Clique Covering Number</a>, <a href="/CliqueNumber.html">Clique Number</a>, <a href="/CliquePolynomial.html">Clique Polynomial</a>, <a href="/LowerCliqueNumber.html">Lower Clique Number</a>, <a href="/MaximalClique.html">Maximal Clique</a>, <a href="/MaximumClique.html">Maximum Clique</a>       <h2>Explore with Wolfram|Alpha</h2> <div id="WAwidget"> <div class="WAwidget-wrapper"> <img alt="WolframAlpha" title="WolframAlpha" src="/images/wolframalpha/WA-logo.png" width="136" height="20"> <form name="wolframalpha" action="https://www.wolframalpha.com/input/" target="_blank"> <input type="text" name="i" class="search" placeholder="Solve your math problems and get step-by-step solutions" value=""> <button type="submit" title="Evaluate on WolframAlpha"></button> </form> </div> <div class="WAwidget-wrapper try"> <p class="text-align-r"> More things to try: </p> <ul> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=%28110110+base+2%29+%2F+%2811+base+2%29">(110110 base 2) / (11 base 2)</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=colorize+image+of+Poe">colorize image of Poe</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=int+sinx%2Fx+dx%2C+x%3D0..infinity">int sinx/x dx, x=0..infinity</a></li> </ul> </div> </div>   <h2>References</h2><cite>Harary, F. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0201410338/ref=nosim/ericstreasuretro">Graph Theory.</a></i> Reading, MA: Addison-Wesley, 1994.</cite><cite>Pemmaraju, S. and Skiena, S. <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521806860/ref=nosim/ericstreasuretro">Computational Discrete Mathematics: Combinatorics and Graph Theory in Mathematica.</a></i> Cambridge, England: Cambridge University Press, pp. 247-248, 2003.</cite><cite>Skiena, S. "Maximum Cliques." §5.6.1 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521806860/ref=nosim/ericstreasuretro">Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.</a></i> Reading, MA: Addison-Wesley, pp. 215 and 217-218, 1990.</cite><cite>Skiena, S. S. "Clique and Independent Set" and "Clique." §6.2.3 and 8.5.1 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0387948600/ref=nosim/ericstreasuretro">The Algorithm Design Manual.</a></i> New York: Springer-Verlag, pp. 144 and 312-314, 1997.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/clique/ls/pi/de/" title="Clique" target="_blank">Clique</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Clique." From <a href="/"><i>MathWorld</i></a>--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/Clique.html">https://mathworld.wolfram.com/Clique.html</a> </p>  <h2>Subject classifications</h2><nav class="breadcrumbs"><ul class="breadcrumb"> <li> <a href="/topics/DiscreteMathematics.html">Discrete Mathematics</a> </li> <li> <a href="/topics/GraphTheory.html">Graph Theory</a> </li> <li> <a href="/topics/Cliques.html">Cliques</a> </li> </ul></nav>  </div> </section> </section>  </div> </main> <aside id="bottom"> <style> #bottom { padding-bottom: 65px; } #acknowledgment { display:none; } .attribution { font-size: .75rem; font-style: italic; } footer ul li:not(:last-of-type)::after { background: #a3a3a3; margin-left: .3rem; margin-right: .1rem; } @media all and (max-width: 900px) { .attribution { font-size: 12px; } } @media (max-width: 600px) { footer { max-width: 360px; } footer ul { max-width: 360px; } footer ul:nth-child(1) li:nth-child(2):after { content: ""; height: 11px; } footer ul:nth-child(1) li:nth-child(3):after { content: ""; height: 0px; } } </style> <footer> <ul> <li><a href="/about/">About MathWorld</a></li> <li><a href="/classroom/">MathWorld Classroom</a></li> <li><a href="/contact/">Contribute</a></li> <li><a href="https://www.amazon.com/exec/obidos/ASIN/1420072218/ref=nosim/weisstein-20" target="_blank">MathWorld Book</a></li> <li class="display-n display-ib__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> </ul> <ul> <li class="display-n__600"><a href="/whatsnew/">13,218 Entries</a></li> <li class="display-n__600"><a href="/whatsnew/">Last Updated: Tue Dec 10 2024</a></li>  <li><a href="https://www.wolfram.com" target="_blank">©1999–2024 Wolfram Research, Inc.</a></li> <li><a href="https://www.wolfram.com/legal/terms/mathworld.html" target="_blank">Terms of Use</a></li> </ul> <ul class="wolfram"> <li class="display-n__600 display-n__900"><a href="https://www.wolfram.com" target="_blank" aria-label="Wolfram"><img src="/images/footer/wolfram-logo.png" alt="Wolfram" title="Wolfram" width="121" height="28"></a></li> <li class="display-n__600"><a href="https://www.wolfram.com" target="_blank">wolfram.com</a></li> <li class="display-n__600"><a href="https://www.wolfram.com/education/" target="_blank">Wolfram for Education</a></li> <li class="attribution">Created, developed and nurtured by Eric Weisstein at Wolfram Research</li> </ul> </footer> <section id="acknowledgment"> <i>Created, developed and nurtured by Eric Weisstein at Wolfram Research</i> </section> </aside> <script type="text/javascript" src="/scripts/scripts.js"></script> <script src="/common/js/c2c/1.0/WolframC2C.js"></script> <script src="/common/js/c2c/1.0/WolframC2CGui.js"></script> <script src="/common/js/c2c/1.0/WolframC2CDefault.js"></script> <link rel="stylesheet" href="/common/js/c2c/1.0/WolframC2CGui.css.en"> <style> .wolfram-c2c-wrapper { padding: 0px !important; border: 0px; } .wolfram-c2c-wrapper:active { border: 0px; } .wolfram-c2c-wrapper:hover { border: 0px; } </style> <script> let c2cWrittings = new WolframC2CDefault({'triggerClass':'mathworld-c2c_above', 'uniqueIdPrefix': 'mathworld-c2c_above-'}); </script> <style> #IPstripe-outer { background: #47a2af; } #IPstripe-outer:hover { background: #0095aa; } </style> <div id="IPstripe-wrap"></div> <script src="/common/stripe/stripe.en.js"></script> </body> </html>