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<!doctype html> <html lang="en" class="discretemathematics historyandterminology"> <head> <title>Tutte Polynomial -- from Wolfram MathWorld</title> <meta name="DC.Title" content="Tutte Polynomial" /> <meta name="DC.Creator" content="Weisstein, Eric W." /> <meta name="DC.Description" content="Let G be an undirected graph, and let i denote the cardinal number of the set of externally active edges of a spanning tree T of G, j denote the cardinal number of the set of internally active edges of T, and t_(ij) the number of spanning trees of G whose internal activity is i and external activity is j. Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by T(x,y)=sumt_(ij)x^iy^j (1) (Biggs 1993, p. 100). An equivalent definition is given by ..." /> <meta name="description" content="Let G be an undirected graph, and let i denote the cardinal number of the set of externally active edges of a spanning tree T of G, j denote the cardinal number of the set of internally active edges of T, and t_(ij) the number of spanning trees of G whose internal activity is i and external activity is j. Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by T(x,y)=sumt_(ij)x^iy^j (1) (Biggs 1993, p. 100). An equivalent definition is given by ..." /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-07-10" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2008-07-27" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2013-11-11" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-03" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-08" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-11" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-05-29" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-06-03" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2014-06-04" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2015-09-30" /> <meta name="DC.Date.Modified" scheme="W3CDTF" content="2017-05-09" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:Discrete Mathematics:Graph Theory:Graph Properties:Graph Polynomials" /> <meta name="DC.Subject" scheme="MathWorld" content="Mathematics:History and Terminology:Wolfram Language Commands" /> <meta name="DC.Rights" content="Copyright 1999-2024 Wolfram Research, Inc. See https://mathworld.wolfram.com/about/terms.html for a full terms of use statement." /> <meta name="DC.Format" scheme="IMT" content="text/html" /> <meta name="DC.Identifier" scheme="URI" content="https://mathworld.wolfram.com/TuttePolynomial.html" /> <meta name="DC.Language" scheme="RFC3066" content="en" /> <meta name="DC.Publisher" content="Wolfram Research, Inc." /> <meta name="DC.Relation.IsPartOf" scheme="URI" content="https://mathworld.wolfram.com/" /> <meta name="DC.Type" scheme="DCMIType" content="Text" /> <meta name="Last-Modified" content="2017-05-09" /> <meta property="og:image" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_TuttePolynomial.png"> <meta property="og:url" content="https://mathworld.wolfram.com/TuttePolynomial.html"> <meta property="og:type" content="website"> <meta property="og:title" content="Tutte Polynomial -- from Wolfram MathWorld"> <meta property="og:description" content="Let G be an undirected graph, and let i denote the cardinal number of the set of externally active edges of a spanning tree T of G, j denote the cardinal number of the set of internally active edges of T, and t_(ij) the number of spanning trees of G whose internal activity is i and external activity is j. Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by T(x,y)=sumt_(ij)x^iy^j (1) (Biggs 1993, p. 100). An equivalent definition is given by ..."> <meta name="twitter:card" content="summary_large_image"> <meta name="twitter:site" content="@WolframResearch"> <meta name="twitter:title" content="Tutte Polynomial -- from Wolfram MathWorld"> <meta name="twitter:description" content="Let G be an undirected graph, and let i denote the cardinal number of the set of externally active edges of a spanning tree T of G, j denote the cardinal number of the set of internally active edges of T, and t_(ij) the number of spanning trees of G whose internal activity is i and external activity is j. Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by T(x,y)=sumt_(ij)x^iy^j (1) (Biggs 1993, p. 100). An equivalent definition is given by ..."> <meta name="twitter:image:src" content="https://mathworld.wolfram.com/images/socialmedia/share/ogimage_TuttePolynomial.png"> <link rel="canonical" href="https://mathworld.wolfram.com/TuttePolynomial.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"> 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href="/topics/HistoryandTerminology.html">History and Terminology</a> </li> <li> <a href="/topics/WolframLanguageCommands.html">Wolfram Language Commands</a> </li> </ul></nav>   <h1>Tutte Polynomial</h1>  <hr class="margin-t-1-8 margin-b-3-4">  <div class="attachments text-align-r"> <a href="/notebooks/GraphTheory/TuttePolynomial.nb" download="TuttePolynomial.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"> <p> Let <img src="/images/equations/TuttePolynomial/Inline1.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> be an <a href="/UndirectedGraph.html">undirected graph</a>, and let <img src="/images/equations/TuttePolynomial/Inline2.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="21" alt="i" /> denote the <a href="/CardinalNumber.html">cardinal number</a> of the set of externally active edges of a spanning tree <img src="/images/equations/TuttePolynomial/Inline3.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" /> of <img src="/images/equations/TuttePolynomial/Inline4.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, <img src="/images/equations/TuttePolynomial/Inline5.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="j" /> denote the <a href="/CardinalNumber.html">cardinal number</a> of the set of internally active edges of <img src="/images/equations/TuttePolynomial/Inline6.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" />, and <img src="/images/equations/TuttePolynomial/Inline7.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="24" alt="t_(ij)" /> the number of spanning trees of <img src="/images/equations/TuttePolynomial/Inline8.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> whose internal activity is <img src="/images/equations/TuttePolynomial/Inline9.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="21" alt="i" /> and external activity is <img src="/images/equations/TuttePolynomial/Inline10.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="j" />. Then the Tutte polynomial, also known as the dichromate or Tutte-Whitney polynomial, is defined by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation1.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="142" height="27" alt=" T(x,y)=sumt_(ij)x^iy^j " /></td><td align="right" width="3"> <div id="eqn1" class="eqnum"> (1) </div> </td></tr> </table> </div> <p> (Biggs 1993, p. 100). </p> <p> An equivalent definition is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation2.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="259" height="27" alt=" T(x,y)=sum(x-1)^(k_A-k)(y-1)^(k_A+n_A-n), " /></td><td align="right" width="3"> <div id="eqn2" class="eqnum"> (2) </div> </td></tr> </table> </div> <p> where the sum is taken over all subsets <img src="/images/equations/TuttePolynomial/Inline11.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" /> of the <a href="/EdgeSet.html">edge set</a> of a graph <img src="/images/equations/TuttePolynomial/Inline12.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, <img src="/images/equations/TuttePolynomial/Inline13.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="k_A" /> is the number of connected components of the subgraph on <img src="/images/equations/TuttePolynomial/Inline14.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="21" alt="n_A" /> vertices induced by <img src="/images/equations/TuttePolynomial/Inline15.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="A" />, <img src="/images/equations/TuttePolynomial/Inline16.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> is the <a href="/VertexCount.html">vertex count</a> of <img src="/images/equations/TuttePolynomial/Inline17.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />, and <img src="/images/equations/TuttePolynomial/Inline18.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="k" /> is the number of connected components of <img src="/images/equations/TuttePolynomial/Inline19.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" />. </p> <p> Several analogs of the Tutte polynomial have been considered for <a href="/DirectedGraph.html">directed graphs</a>, including the cover polynomial (Chung and Graham 1995), Gordon-Traldi polynomials (Gordon and Traldi 1993), and three-variable <img src="/images/equations/TuttePolynomial/Inline20.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="B" />-polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of the the Gordon-Traldi polynomial <img src="/images/equations/TuttePolynomial/Inline21.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="22" alt="f_8" /> and <img src="/images/equations/TuttePolynomial/Inline22.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="11" height="21" alt="B" />-polynomial, these are not proper generalizations of the Tutte polynomial since they are not equivalent to the Tutte polynomial for the special case of undirected graphs (Awan and Bernardi 2016). </p> <p> The Tutte polynomial can be computed in the <a href="http://www.wolfram.com/language/">Wolfram Language</a> using <tt><a href="http://reference.wolfram.com/language/ref/TuttePolynomial.html">TuttePolynomial</a></tt>[<i>g</i>, <img src="/images/equations/TuttePolynomial/Inline23.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="{" /><i>x</i>, <i>y</i><img src="/images/equations/TuttePolynomial/Inline24.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="6" height="21" alt="}" />]. </p> <p> The Tutte polynomial is multiplicative over disjoint unions. </p> <p> For an <a href="/UndirectedGraph.html">undirected graph</a> on <img src="/images/equations/TuttePolynomial/Inline25.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> vertices with <img src="/images/equations/TuttePolynomial/Inline26.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="c" /> connected components, the Tutte polynomial is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation3.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="215" height="23" alt=" T(x+1,y+1)=x^(n-c)R(x^(-1),y) " /></td><td align="right" width="3"> <div id="eqn3" class="eqnum"> (3) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/TuttePolynomial/Inline27.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="R(x,y)" /> is the <a href="/RankPolynomial.html">rank polynomial</a> (generalizing Biggs 1993, p. 101). The Tutte polynomial is therefore a rather general two-variable graph polynomial from which a number of other important one- and two-variable polynomials can be computed. </p> <p> For not-necessarily connected graphs, the Tutte polynomial <img src="/images/equations/TuttePolynomial/Inline28.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="T(x,y)" /> is related the <a href="/ChromaticPolynomial.html">chromatic polynomial</a> <img src="/images/equations/TuttePolynomial/Inline29.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="pi(x)" />, <a href="/FlowPolynomial.html">flow polynomial</a> <img src="/images/equations/TuttePolynomial/Inline30.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="43" height="21" alt="C^*(u)" />, <a href="/RankPolynomial.html">rank polynomial</a> <img src="/images/equations/TuttePolynomial/Inline31.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="21" alt="R(x,y)" />, and <a href="/ReliabilityPolynomial.html">reliability polynomial</a> <img src="/images/equations/TuttePolynomial/Inline32.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="C(p)" /> by </p> <div class="table-responsive-noborders"> <table align="center" width="100%" cellpadding="0" cellspacing="0" style="padding-left: 50px" border="0"> <tr style=""><td align="right" width=""><img src="/images/equations/TuttePolynomial/Inline33.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="29" height="20" alt="pi(x)" /></td><td align="center" width="14"><img src="/images/equations/TuttePolynomial/Inline34.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline35.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="20" alt="(-1)^(n-c)x^cT(1-x,0)" /></td><td align="right" width="10"> <div id="eqn4" class="eqnum"> (4) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/TuttePolynomial/Inline36.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="40" height="20" alt="C^*(u)" /></td><td align="center" width="14"><img src="/images/equations/TuttePolynomial/Inline37.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline38.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="20" alt="(-1)^(m-n+c)T(0,1-u)" /></td><td align="right" width="10"> <div id="eqn5" class="eqnum"> (5) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/TuttePolynomial/Inline39.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="50" height="20" alt="R(x,y)" /></td><td align="center" width="14"><img src="/images/equations/TuttePolynomial/Inline40.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline41.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="144" height="23" alt="x^(n-c)T(x^(-1)+1,y+1)" /></td><td align="right" width="10"> <div id="eqn6" class="eqnum"> (6) </div> </td></tr><tr style=""><td align="right" width=""><img src="/images/equations/TuttePolynomial/Inline42.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="35" height="20" alt="C(p)" /></td><td align="center" width="14"><img src="/images/equations/TuttePolynomial/Inline43.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="10" height="20" alt="=" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline44.svg" class="displayformula" style="max-height:100%;max-width:100%" border="0" width="185" height="23" alt="(1-p)^(n-c)p^(m-n+c)T(1,p^(-1))," /></td><td align="right" width="10"> <div id="eqn7" class="eqnum"> (7) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/TuttePolynomial/Inline45.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> is the number of vertices in the graph, <img src="/images/equations/TuttePolynomial/Inline46.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="m" /> is the number of edges, and <img src="/images/equations/TuttePolynomial/Inline47.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="c" /> is the number of connected components. </p> <p> The Tutte polynomial of the <a href="/DualGraph.html">dual graph</a> <img src="/images/equations/TuttePolynomial/Inline48.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="G^*" /> of a graph <img src="/images/equations/TuttePolynomial/Inline49.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation4.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="146" height="21" alt=" T_(G^*)(x,y)=T_G(y,x), " /></td><td align="right" width="3"> <div id="eqn8" class="eqnum"> (8) </div> </td></tr> </table> </div> <p> i.e., by swapping the variables of the Tutte polynomial of the original graph. A special case of this identity relates the <a href="/FlowPolynomial.html">flow polynomial</a> <img src="/images/equations/TuttePolynomial/Inline50.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="45" height="23" alt="C_G^*(u)" /> of a <a href="/PlanarGraph.html">planar graph</a> <img src="/images/equations/TuttePolynomial/Inline51.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> to the <a href="/ChromaticPolynomial.html">chromatic polynomial</a> of its <a href="/DualGraph.html">dual graph</a> <img src="/images/equations/TuttePolynomial/Inline52.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="G^*" /> by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation5.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="139" height="22" alt=" C_G^*(u)=u^(-1)pi_(G^*)(u). " /></td><td align="right" width="3"> <div id="eqn9" class="eqnum"> (9) </div> </td></tr> </table> </div> <p> The Tutte polynomial of a connected graph <img src="/images/equations/TuttePolynomial/Inline53.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> is also completely defined by the following two properties (Biggs 1993, p. 103): </p> <p> 1. If <img src="/images/equations/TuttePolynomial/Inline54.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="e" /> is an edge of <img src="/images/equations/TuttePolynomial/Inline55.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> which is neither a loop nor an isthmus, then <img src="/images/equations/TuttePolynomial/Inline56.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="284" height="25" alt="T_G(x,y)=T(G^((e));x,y)+T(G_((e));x,y)" />. </p> <p> 2. If <img src="/images/equations/TuttePolynomial/Inline57.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="26" height="24" alt="Lambda_(ij)" /> is formed from a tree with <img src="/images/equations/TuttePolynomial/Inline58.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="5" height="21" alt="i" /> edges by adding <img src="/images/equations/TuttePolynomial/Inline59.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="8" height="21" alt="j" /> loops, then <img src="/images/equations/TuttePolynomial/Inline60.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="144" height="25" alt="T(Lambda_(ij);x,y)=x^iy^j" /> </p> <p> Closed forms for some special classes of graphs are summarized in the following table, where <img src="/images/equations/TuttePolynomial/Inline61.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="227" height="35" alt="s=sqrt((1+x+x^2+y)^2-4x^2y)" /> and <img src="/images/equations/TuttePolynomial/Inline62.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="182" height="28" alt="t=sqrt((x+y+1)^2-4xy)" />. The Tutte polynomial of the <a href="/WebGraph.html">web graph</a> was considered by Biggs <i>et al. </i>(1972) and Brennan <i>et al. </i>(2013). </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline63.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="54" height="21" alt="T(x,y)" /></td></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/TuttePolynomial/Inline64.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/TuttePolynomial/Inline65.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="197" height="34" alt="((1+x+x^2)^n[x(y-1)-y]+y(x+x^2+y)^n)/(y-1)" /></td></tr><tr style=""><td align="left"><a href="/CentipedeGraph.html">centipede graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline66.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="x^(2n-1)" /></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/TuttePolynomial/Inline67.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/TuttePolynomial/Inline68.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="60" height="27" alt="(x^n-x)/(x-1)+y" /></td></tr><tr style=""><td align="left"><a href="/EmptyGraph.html">empty graph</a> <img src="/images/equations/TuttePolynomial/Inline69.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="K^__n" /></td><td align="left">1</td></tr><tr style=""><td align="left"><a href="/Forest.html">forest</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline70.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="24" height="21" alt="x^(|E|)" /></td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline71.svg" data-src-small="/images/equations/TuttePolynomial/Inline71_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline71.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="542" height="25" data-big="542 25" data-small="349 54" border="0" alt="2^(-n)(-2^n-2^nx-2^ny+2^nxy+(1+x+x^2+y-s)^n+(1+x+x^2+y+s)^n" /></td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline72.svg" data-src-small="/images/equations/TuttePolynomial/Inline72_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline72.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="440" height="22" data-big="440 22" data-small="287 47" border="0" alt="x^n{-1-x-y+xy+2^(-n)[(1-t+x+y)^n+(1+t+x+y)^n]}" /></td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline73.svg" data-src-small="/images/equations/TuttePolynomial/Inline73_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline73.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="425" height="34" data-big="425 34" data-small="374 88" border="0" alt="((s(x-1)+x^3-xy+y+1)(s+x^2+x+y+1)^n-(-sx+s+x^3-xy+y+1)(-s+x^2+x+y+1)^n)/(2^(n+1)sx^2)" /></td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline74.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^n" /></td></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline75.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="96" height="28" alt="(x[x^n+x(y-1)-y])/(x-1)" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a> <img src="/images/equations/TuttePolynomial/Inline76.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/TuttePolynomial/Inline77.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="x^(n-1)" /></td></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/TuttePolynomial/Inline78.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/TuttePolynomial/Inline79.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="30" height="21" alt="x^(n-1)" /></td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/TuttePolynomial/Inline80.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/TuttePolynomial/Inline81.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="117" height="30" alt="(x^(2n)+x^n(x(-1+y)-y))/(x-1)" /></td></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/TuttePolynomial/Inline82.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/TuttePolynomial/Inline83.svg" data-src-small="/images/equations/TuttePolynomial/Inline83_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline83.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="430" height="25" data-big="430 25" data-small="280 53" border="0" alt="xy-x-y-1+2^(1-n)[(x+y+1+t)^(n-1)+(x+y+1-t)^(n-1)]" /></td></tr> </table> </div> <p> The following table summarizes the recurrence relations for Tutte polynomials for some simple classes of graphs. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="left">graph</td><td align="right">order</td><td align="left">recurrence</td></tr><tr style=""><td align="left"><a href="/AntiprismGraph.html">antiprism graph</a></td><td align="right">6</td><td align="left" /></tr><tr style=""><td align="left"><a href="/BookGraph.html">book graph</a> <img src="/images/equations/TuttePolynomial/Inline84.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="53" height="22" alt="S_(n+1)P_2" /></td><td align="right">2</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline85.svg" data-src-small="/images/equations/TuttePolynomial/Inline85_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline85.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="434" height="25" data-big="434 25" data-small="270 54" border="0" alt="p_n=(2x^2+2x+y+1)p_(n-1)-(x^2+x+1)(x^2+x+y)p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/CentipedeGraph.html">centipede graph</a></td><td align="right">1</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline86.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="91" height="22" alt="p_n=x^2p_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/CycleGraph.html">cycle graph</a> <img src="/images/equations/TuttePolynomial/Inline87.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="C_n" /></td><td align="right">2</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline88.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="186" height="22" alt="p_n=(x+1)p_(n-1)-xp_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/GearGraph.html">gear graph</a></td><td align="right">3</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline89.svg" data-src-small="/images/equations/TuttePolynomial/Inline89_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline89.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="494" height="25" data-big="494 25" data-small="305 54" border="0" alt="p_n=(x^2+x+y+2)p_(n-1)+(-x^2y-x^2-x-y-1)p_(n-2)+x^2yp_(n-3)" /></td></tr><tr style=""><td align="left"><a href="/HelmGraph.html">helm graph</a></td><td align="right">3</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline90.svg" data-src-small="/images/equations/TuttePolynomial/Inline90_400.svg" data-src-default="/images/equations/TuttePolynomial/Inline90.svg" class="inlineformula swappable" style="max-height:100%;max-width:100%" width="407" height="22" data-big="407 22" data-small="240 48" border="0" alt="p_n=x(x+y-2)p_(n-1)-x^2(x+1)(y+1)p_(n-2)+x^4p_(n-3)" /></td></tr><tr style=""><td align="left"><a href="/LadderGraph.html">ladder graph</a></td><td align="right">2</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline91.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="269" height="25" alt="p_n=(x^2+x+y+1)p_(n-1)-x^2yp_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/LadderRungGraph.html">ladder rung graph</a></td><td align="right">1</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline92.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="84" height="22" alt="p_n=xp_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/MoebiusLadder.html">Möbius ladder</a> <img src="/images/equations/TuttePolynomial/Inline93.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="M_n" /></td><td align="right">6</td><td align="left" /></tr><tr style=""><td align="left"><a href="/PanGraph.html">pan graph</a></td><td align="right">2</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline94.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="186" height="22" alt="p_n=(x+1)p_(n-1)-xp_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/PathGraph.html">path graph</a></td><td align="right">1</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline95.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="84" height="22" alt="p_n=xp_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/PrismGraph.html">prism graph</a> <img src="/images/equations/TuttePolynomial/Inline96.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="17" height="21" alt="Y_n" /></td><td align="right">6</td><td align="left" /></tr><tr style=""><td align="left"><a href="/StarGraph.html">star graph</a> <img src="/images/equations/TuttePolynomial/Inline97.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="S_n" /></td><td align="right">1</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline98.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="84" height="22" alt="p_n=xp_(n-1)" /></td></tr><tr style=""><td align="left"><a href="/SunletGraph.html">sunlet graph</a> <img src="/images/equations/TuttePolynomial/Inline99.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="right">2</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline100.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="206" height="22" alt="p_n=x(x+1)p_(n-1)-x^3p_(n-2)" /></td></tr><tr style=""><td align="left"><a href="/WebGraph.html">web graph</a></td><td align="right">6</td><td align="left" /></tr><tr style=""><td align="left"><a href="/WheelGraph.html">wheel graph</a> <img src="/images/equations/TuttePolynomial/Inline101.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="22" height="21" alt="W_n" /></td><td align="right">3</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline102.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="380" height="22" alt="p_n=(x+y+2)p_(n-1)-(x+1)(y+1)p_(n-2)+xyp_(n-3)" /></td></tr> </table> </div> <p> An equation for the Tutte polynomial <img src="/images/equations/TuttePolynomial/Inline103.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="68" height="24" alt="T_(K_n)(x,y)" /> of the <a href="/CompleteGraph.html">complete graph</a> <img src="/images/equations/TuttePolynomial/Inline104.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /> was found by Tutte (1954, 1967). In particular, <img src="/images/equations/TuttePolynomial/Inline105.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="68" height="24" alt="T_(K_n)(x,y)" /> has <a href="/ExponentialGeneratingFunction.html">exponential generating function</a> </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation6.svg" data-src-small="/images/equations/TuttePolynomial/NumberedEquation6_400.svg" data-src-default="/images/equations/TuttePolynomial/NumberedEquation6.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="425" height="56" data-big="425 56" data-small="264 116" border="0" alt=" sum_(n=1)^inftyT_(K_n)(x,y)(u^n)/(n!)=1/(x-1){[sum_(n=0)^inftyy^((n; 2))(y-1)^(-n)(u^n)/(n!)]^((x-1)(y-1))-1}, " /></td><td align="right" width="3"> <div id="eqn10" class="eqnum"> (10) </div> </td></tr> </table> </div> <p> (Gessel 1995, Gessel and Sagan 1996). This can be written more simply in terms of the coboundary polynomial </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation7.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="269" height="40" alt=" chi^__G(q,t)=(t-1)^(n_G-c_G)T_G((q+t-1)/(t-1),t), " /></td><td align="right" width="3"> <div id="eqn11" class="eqnum"> (11) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/TuttePolynomial/Inline106.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="18" height="22" alt="c_G" /> is the connected component count and <img src="/images/equations/TuttePolynomial/Inline107.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="n_G" /> is the <a href="/VertexCount.html">vertex count</a> of a graph <img src="/images/equations/TuttePolynomial/Inline108.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="13" height="21" alt="G" /> (Martin and Reiner 2005). In this form, the exponential generating function of <img src="/images/equations/TuttePolynomial/Inline109.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="20" height="21" alt="K_n" /> is given by </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation8.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="261" height="53" alt=" 1+qsum_(n=1)^inftychi^__(K_n)(q,t)(x^n)/(n!)=(sum_(n=0)^inftyt^((n; 2))(x^n)/(n!))^q, " /></td><td align="right" width="3"> <div id="eqn12" class="eqnum"> (12) </div> </td></tr> </table> </div> <p> which can be converted to the corresponding Tutte polynomial using the above relationship and the substitution <img src="/images/equations/TuttePolynomial/Inline110.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="129" height="21" alt="q->(x-1)(y-1)" /> and <img src="/images/equations/TuttePolynomial/Inline111.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="36" height="21" alt="t->y" />. The formula was rediscovered by Pak in the form of the following recurrence </p> <div> <table summary="" width="100%" align="center" cellspacing="0" cellpadding="0" style="padding-left: 50px"> <tr><td align="left"><img src="/images/equations/TuttePolynomial/NumberedEquation9.svg" data-src-small="/images/equations/TuttePolynomial/NumberedEquation9_400.svg" data-src-default="/images/equations/TuttePolynomial/NumberedEquation9.svg" class="numberedequation swappable" style="max-height:100%;max-width:100%" width="460" height="51" data-big="460 51" data-small="317 83" border="0" alt=" F_n(x,y)=sum_(k=1)^n(n-1; k-1)(x+y+y^2+...+y^(k-1))F_(k-1)(1,y)F_(n-k)(x,y), " /></td><td align="right" width="3"> <div id="eqn13" class="eqnum"> (13) </div> </td></tr> </table> </div> <p> where <img src="/images/equations/TuttePolynomial/Inline112.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="159" height="24" alt="F_n(x,y)=T_(K_(n+1))(x,y)" />. </p> <p> A formula for the Tutte polynomial of a <a href="/CompleteBipartiteGraph.html">complete bipartite graph</a> <img src="/images/equations/TuttePolynomial/Inline113.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="23" alt="K_(m,n)" /> is given in terms of an bivariate <a href="/ExponentialGeneratingFunction.html">exponential generating function</a> for the coboundary polynomial 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/TuttePolynomial/NumberedEquation10.svg" class="numberedequation" style="max-height:100%;max-width:100%" border="0" width="352" height="53" alt=" 1+qsum_(m=0)^inftysum_(n=0)^inftychi^__(K_(m,n))(q,t)(x^my^m)/(m!n!)=(sum_(k=0)^inftysum_(l=0)^inftyt^(kl)(x^ky^l)/(k!l!))^q " /></td><td align="right" width="3"> <div id="eqn14" class="eqnum"> (14) </div> </td></tr> </table> </div> <p> by Martin and Reiner (2005). </p> <p> Nonisomorphic graphs do not necessarily have distinct Tutte polynomials. de Mier and Noy (2004) call a graph that is determined by its Tutte polynomial a <img src="/images/equations/TuttePolynomial/Inline114.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" />-unique graph and showed that <a href="/WheelGraph.html">wheel graphs</a>, <a href="/LadderGraph.html">ladder graphs</a>, <a href="/MoebiusLadder.html">Möbius ladders</a>, complete multipartite graphs (with the exception of <img src="/images/equations/TuttePolynomial/Inline115.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="28" height="24" alt="T_(1,p)" />), and <a href="/HypercubeGraph.html">hypercube graphs</a> are <img src="/images/equations/TuttePolynomial/Inline116.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" />-unique graphs. Kuhl (2008) showed that the <a href="/GeneralizedPetersenGraph.html">generalized Petersen graphs</a> <img src="/images/equations/TuttePolynomial/Inline117.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="73" height="21" alt="GP(m,2)" /> and their <a href="/LineGraph.html">line graphs</a> <img src="/images/equations/TuttePolynomial/Inline118.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="98" height="21" alt="L(GP(m,2))" /> are <img src="/images/equations/TuttePolynomial/Inline119.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="12" height="21" alt="T" />-unique. </p> <p> The numbers of simple graphs on <img src="/images/equations/TuttePolynomial/Inline120.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="39" height="21" alt="n=1" />, 2, ... nodes that are not Tutte-unique for a given value of <img src="/images/equations/TuttePolynomial/Inline121.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /> are 0, 0, 0, 4, 15, 84, 548, 5629, ... (OEIS <a href="http://oeis.org/A243048">A243048</a>), while the corresponding numbers of Tutte-unique graphs are 1, 2, 4, 7, 19, 72, 496, 6717, ... (OEIS <a href="http://oeis.org/A243049">A243049</a>). The following table summarizes some small co-Tutte graphs. </p> <div class="table-responsive"> <table align="center" class="mathworldtable"> <tr style=""><td align="right"><img src="/images/equations/TuttePolynomial/Inline122.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="9" height="21" alt="n" /></td><td align="left">Tutte polynomial</td><td align="left">graphs</td></tr><tr style=""><td align="right">4</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline123.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^2" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline124.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="62" height="22" alt="P_3 union K_1" />, ladder rung graph <img src="/images/equations/TuttePolynomial/Inline125.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="32" height="21" alt="2P_2" /></td></tr><tr style=""><td align="right">4</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline126.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^3" /></td><td align="left"><a href="/ClawGraph.html">claw graph</a> <img src="/images/equations/TuttePolynomial/Inline127.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="29" height="23" alt="K_(1,3)" />, <a href="/PathGraph.html">path graph</a> <img src="/images/equations/TuttePolynomial/Inline128.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="21" alt="P_4" /></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline129.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^2" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline130.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="75" height="22" alt="P_3 union 2K_1" />, <img src="/images/equations/TuttePolynomial/Inline131.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="75" height="21" alt="2P_2 union K_1" /></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline132.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^3" /></td><td align="left"><img src="/images/equations/TuttePolynomial/Inline133.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="72" height="23" alt="K_(1,3) union K_1" />, <img src="/images/equations/TuttePolynomial/Inline134.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="22" alt="P_3 union P_2" />, <img src="/images/equations/TuttePolynomial/Inline135.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="21" alt="P_4 union P_1" /></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline136.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="21" alt="x^4" /></td><td align="left"><a href="/ForkGraph.html">fork graph</a>, <a href="/PathGraph.html">path graph</a> <img src="/images/equations/TuttePolynomial/Inline137.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="19" height="22" alt="P_5" />, <a href="/StarGraph.html">star graph</a> <img src="/images/equations/TuttePolynomial/Inline138.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="16" height="22" alt="S_5" /></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline139.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="95" height="25" alt="x(x+x^2+y)" /></td><td align="left"><a href="/PawGraph.html">paw graph</a> <img src="/images/equations/TuttePolynomial/Inline140.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="35" height="21" alt=" union K_1" />, <img src="/images/equations/TuttePolynomial/Inline141.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="61" height="22" alt="C_3 union P_2" /></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline142.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="102" height="25" alt="x^2(x+x^2+y)" /></td><td align="left"><a href="/BullGraph.html">bull graph</a>, <a href="/CricketGraph.html">cricket graph</a>, <img src="/images/equations/TuttePolynomial/Inline143.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="38" height="21" alt="(3,2)" />-<a href="/TadpoleGraph.html">tadpole graph</a></td></tr><tr style=""><td align="right">5</td><td align="left"><img src="/images/equations/TuttePolynomial/Inline144.svg" class="inlineformula" style="max-height:100%;max-width:100%" border="0" width="232" height="25" alt="x(x+2x^2+x^3+y+2xy+y^2)" /></td><td align="left"><a href="/DartGraph.html">dart graph</a>, <a href="/KiteGraph.html">kite graph</a></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="/ChromaticPolynomial.html">Chromatic Polynomial</a>, <a href="/FlowPolynomial.html">Flow Polynomial</a>, <a href="/RankPolynomial.html">Rank Polynomial</a>, <a href="/TutteMatrix.html">Tutte Matrix</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=ANF+%28%7EP+%7C%7C+Q%29+%26%26+%28P+%7C%7C+%7EQ%29">ANF (~P || Q) && (P || ~Q)</a></li> <li><a target="_blank" href="https://www.wolframalpha.com/input/?i=continued+fraction+12%2F67">continued fraction 12/67</a></li> <li><a target="_blank" href="http://www.wolframalpha.com/input/?i=horizontal+asymptotes+tanh%28x%5E2%29">horizontal asymptotes tanh(x^2)</a></li> </ul> </div> </div>   <h2>References</h2><cite>Andrzejak, A. "Splitting Formulas for Tutte Polynomials." <i>J. Combin. Th., Ser. B.</i> <b>70</b>, 346-366, 1997.</cite><cite>Andrzejak, A. "An Algorithm for the Tutte Polynomials of Graphs of Bounded Treewidth." <i>Disc. Math.</i> <b>190</b>, 39-54, 1998.</cite><cite>Biggs, N. L. "The Tutte Polynomial." Ch. 13 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521458978/ref=nosim/ericstreasuretro">Algebraic Graph Theory, 2nd ed.</a></i> Cambridge, England: Cambridge University Press, pp. 97-105, 1993.</cite><cite>Biggs, N. L.; Damerell, R. M.; and Sands, D. A. "Recursive Families of Graphs." <i>J. Combin. Theory Ser. B</i> <b>12</b>, 123-131, 1972.</cite><cite>Björklund, A.; Husfeldt, T.; Kaski, P.; and Koivisto, M. "Computing the Tutte Polynomial in Vertex-Exponential Time." In <i>Proceedings of the IEEE Symposium on the Foundations of Computer Science (FOCS)</i>, 677-686, 2008.</cite><cite>Brennan, C.; Mansour, T.; and Mphako-Banda, E. "Tutte Polynomials of Wheels Via Generating Functions." <i>Bull. Iranian Math. Soc.</i> <b>39</b>, 881-891, 2013.</cite><cite>Brylawski, T. and Oxley, J. "The Tutte Polynomial and Its Applications." Ch. 6 in <i><a href="http://www.amazon.com/exec/obidos/ASIN/0521381657/ref=nosim/ericstreasuretro">Matroid Applications</a></i> (Ed. N. White). Cambridge, England: Cambridge University Press, pp. 123-225, 1992.</cite><cite>Chow, T Y. "Digraph Analogues of the Tutte Polynomials." Preprint chapter for <i>The CRC Handbook on the Tutte Polynomial and Related Topics</i> (Ed. I. Moffat and J. Ellis-Monaghan). 2016.</cite><cite>Chung, F. R. K. and Graham, R. L. "On the Cover Polynomial of a Digraph." <i>J. Combin. Theory, Ser. B</i> <b>65</b>, 273-290, 1995.</cite><cite>de Mier, A. and Noy, M. "On Graphs Determined by Their Tutte Polynomial." <i>Graphs Combin.</i> <b>20</b>, 105-119, 2004.</cite><cite>de Mier, A. and Noy, M. "Tutte Uniqueness of Line Graphs." <i>Disc. Math.</i> <b>301</b>, 57-65, 2005.</cite><cite>Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications I: The Tutte Polynomial." 28 Jun 2008. <a href="http://arxiv.org/abs/0803.3079">http://arxiv.org/abs/0803.3079</a>.</cite><cite>Ellis-Monaghan, J. A. and Merino, C. "Graph Polynomials and Their Applications II: Interrelations and Interpretations." 28 Jun 2008. <a href="http://arxiv.org/abs/0806.4699">http://arxiv.org/abs/0806.4699</a>.</cite><cite>Gessel, I. M. "Enumerative Applications of a Decomposition for Graphs and Digraphs." <i>Disc. Math.</i> <b>139</b>, 257-271, 1995.</cite><cite>Gessel, I. M. and Sagan, B. E. "The Tutte Polynomial of a Graph, Depth-First Search, and Simplicial Complex Partitions." <i>Electronic J. Combinatorics</i> <b>3</b>, No. 2, R9, 1-36, 1996. <a href="http://www.combinatorics.org/Volume_3/Abstracts/v3i2r9.html">http://www.combinatorics.org/Volume_3/Abstracts/v3i2r9.html</a>.</cite><cite>Gordon, G. and Traldi, L. "Polynomials for Directed Graphs." <i>Congr. Numer.</i> <b>94</b>, 187-201, 1993.</cite><cite>Haggard, G.; Pearce, D. J.; and Royle, G. "Computing Tutte Polynomials" <i>ACM Trans. Math. Software</i> <b>37</b>, Art. 24, 17 pp., 2010.</cite><cite>Jaeger, F. "Tutte Polynomials and Link Polynomials." <i>Proc. Amer. Math. Soc.</i> <b>103</b>, 647-665, 1988.</cite><cite>Jaeger, F.; Vertigan, D.; and Welsh, D. "On the Computational Complexity of the Jones and Tutte Polynomials." <i>Math. Proc. Camb. Phil. Soc.</i> <b>108</b>, 35-53, 1990.</cite><cite>Kuhl, J. S. "The Tutte Polynomial and the Generalized Petersen Graph." <i>Australas. J. Combin.</i> <b>40</b>, 87-97, 2008.</cite><cite>Pak, I. "Computation of Tutte Polynomials of Complete Graphs." <a href="http://www.math.ucla.edu/~pak/papers/Pak_Computation_Tutte_polynomial_complete_graphs.pdf">http://www.math.ucla.edu/~pak/papers/Pak_Computation_Tutte_polynomial_complete_graphs.pdf</a>.</cite><cite>Martin, J. and Reiner, V. "Cyclotomic and Simplicial Matroids." <i>Israel J. Math.</i> <b>150</b>, 229-240, 2005.</cite><cite>Sloane, N. J. A. Sequences <a href="http://oeis.org/A243048">A243048</a> and <a href="http://oeis.org/A243049">A243049</a> in "The On-Line Encyclopedia of Integer Sequences."</cite><cite>Tutte, W. T. "A Contribution to the Theory of Chromatic Polynomials." <i>Canad. J. Math.</i> <b>6</b>, 80-91, 1954.</cite><cite>Tutte, W. T. "On Dichromatic Polynomials." <i>J. Combin. Th.</i> <b>2</b>, 301-320, 1967.</cite><h2>Referenced on Wolfram|Alpha</h2><a href="http://www.wolframalpha.com/entities/mathworld/tutte_polynomial/ng/en/e5/" title="Tutte Polynomial" target="_blank">Tutte Polynomial</a>   <h2>Cite this as:</h2> <p> <a href="/about/author.html">Weisstein, Eric W.</a> "Tutte Polynomial." 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