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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202101.png" /> be a [[Matroid|matroid]] with rank function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202102.png" /> on the ground set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202103.png" />. The Tutte polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202105.png" /> is defined by
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the semi-automatic procedure described at https://encyclopediaofmath.org/wiki/User:Maximilian_Janisch/latexlist
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If the TeX and formula formatting is correct and if all png images have been replaced by TeX code, please remove this message and the {{TEX|semi-auto}} category.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202106.png" /></td> </tr></table>
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Out of 94 formulas, 91 were replaced by TEX code.-->
  
One also writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202107.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202108.png" /> when no confusion can arise about the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t1202109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021010.png" />. There are several equivalent reformulations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021011.png" />; the most useful such expression is recursive, using the matroid operations of deletion and contraction:
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Let $M$ be a [[Matroid|matroid]] with rank function $r$ on the ground set $E$. The Tutte polynomial $t ( M ; x , y )$ of $M$ is defined by
  
T0) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021013.png" />;
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\begin{equation*} t ( M ; x , y ) = \sum _ { S \subseteq E } ( x - 1 ) ^ { r ( M ) - r ( S ) } ( y - 1 ) ^ { | S |  - r ( S )}. \end{equation*}
  
T1) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021014.png" /> is an isthmus, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021015.png" />;
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One also writes $t ( M )$ for $t ( M ; x , y )$ when no confusion can arise about the values of $x$ and $y$. There are several equivalent reformulations of $t ( M )$; the most useful such expression is recursive, using the matroid operations of deletion and contraction:
  
T2) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021016.png" /> is a loop, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021017.png" />;
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T0) If $E = \emptyset$, then $t ( M ) = 1$;
  
T3) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021018.png" /> is neither an isthmus nor a loop, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021019.png" />.
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T1) If $e$ is an isthmus, then $t ( M ) = x t ( M / e )$;
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T2) If $e$ is a loop, then $t ( M ) = y t ( M - e )$;
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T3) If $e$ is neither an isthmus nor a loop, then $t ( M ) = t ( M / e ) + t ( M - e )$.
  
 
Some standard evaluations of the Tutte polynomial are:
 
Some standard evaluations of the Tutte polynomial are:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021020.png" /> is the number of bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021021.png" />;
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a) $t ( M ; 1,1 )$ is the number of bases of $M$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021022.png" /> is the number of independent subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021023.png" />;
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b) $t ( M ; 2,1 )$ is the number of independent subsets of $M$;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021024.png" /> is the number of spanning subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021025.png" />;
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c) $t ( M ; 1,2 )$ is the number of spanning subsets of $M$;
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021026.png" />.
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d) $t ( M ; 2,2 ) = 2 ^ { | E | }$.
  
When a matroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021027.png" /> is constructed from other matroids <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021028.png" />, it is frequently possible to compute <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021029.png" /> from the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021030.png" />. The most fundamental structural result of this kind concerns the direct sum of matroids: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021031.png" />. Another fundamental structural property of the Tutte polynomial is its transparent relationship with matroid duality: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021032.png" />. Many related results can be found in [[#References|[a3]]].
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When a matroid $M$ is constructed from other matroids $M _ { 1 } , M _ { 2 } , \ldots$, it is frequently possible to compute $t ( M )$ from the $t ( M _ { i } )$. The most fundamental structural result of this kind concerns the direct sum of matroids: $t ( M _ { 1 } \oplus M _ { 2 } ) = t ( M _ { 1 } ) t ( M _ { 2 } )$. Another fundamental structural property of the Tutte polynomial is its transparent relationship with matroid duality: $t ( M ^ { * } ; x , y ) = t ( M ; y , x )$. Many related results can be found in [[#References|[a3]]].
  
The Tutte polynomial has many significant connections with invariants in [[Graph theory|graph theory]]. Of central importance is the relationship between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021033.png" /> and the chromatic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021034.png" /> of a graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021035.png" /> (cf. also [[Graph colouring|Graph colouring]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021036.png" /> is a graph having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021037.png" /> vertices and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021038.png" /> connected components, then
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The Tutte polynomial has many significant connections with invariants in [[Graph theory|graph theory]]. Of central importance is the relationship between $t ( M )$ and the [[chromatic polynomial]] $\chi ( G ; \lambda )$ of a graph $G$ (cf. also [[Graph colouring|Graph colouring]]). If $G$ is a graph having $v ( G )$ vertices and $c ( G )$ connected components, then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021039.png" /></td> </tr></table>
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\begin{equation*} \chi ( G ; \lambda ) = \lambda ^ { c ( G ) } ( - 1 ) ^ { v ( G ) - c ( G ) } t ( M _ { G } , 1 - \lambda , 0 ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021040.png" /> is the cycle matroid of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021041.png" /> (cf. also [[Matroid|Matroid]]). Thus, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021042.png" /> is planar (cf. [[Graph, planar|Graph, planar]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021043.png" /> simultaneously carries information concerning proper colourings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021044.png" /> along with proper colourings of its dual graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021045.png" />. This is the reason W.T. Tutte referred to it as a dichromatic polynomial in his foundational work in this area [[#References|[a14]]]. The polynomial was generalized from graphs to matroids by H.H. Crapo [[#References|[a5]]] and T.H. Brylawski [[#References|[a2]]].
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where $M _ { G }$ is the cycle matroid of $G$ (cf. also [[Matroid|Matroid]]). Thus, when $G$ is planar (cf. [[Graph, planar|Graph, planar]]), $t ( M _ { G } ; x , y )$ simultaneously carries information concerning proper colourings of $G$ along with proper colourings of its [[dual graph]] $G ^ { * }$. This is the reason W.T. Tutte referred to it as a dichromatic polynomial in his foundational work in this area [[#References|[a14]]]. The polynomial was generalized from graphs to matroids by H.H. Crapo [[#References|[a5]]] and T.H. Brylawski [[#References|[a2]]].
  
Other invariants which can be obtained from the Tutte polynomial include the beta invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021046.png" />, the Möbius function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021047.png" />, and the characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021048.png" /> of a matroid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021049.png" />. See [[#References|[a16]]] for more information about these invariants.
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Other invariants which can be obtained from the Tutte polynomial include the beta invariant $\beta ( M )$, the Möbius function $\mu ( M )$, and the characteristic polynomial $p ( M ; \lambda )$ of a matroid $M$. See [[#References|[a16]]] for more information about these invariants.
  
 
==Applications==
 
==Applications==
 
The Tutte polynomial has applications in various areas, some of which are given below. For an extensive introduction, see [[#References|[a4]]].
 
The Tutte polynomial has applications in various areas, some of which are given below. For an extensive introduction, see [[#References|[a4]]].
  
[[Acyclic orientation]]s: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021050.png" /> denote the number of acyclic orientations of a graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021051.png" />. Then R. Stanley [[#References|[a13]]] proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021052.png" />. A variety of related evaluations appear in [[#References|[a9]]].
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[[Acyclic orientation]]s: Let $a ( G )$ denote the number of acyclic orientations of a graph $G$. Then R. Stanley [[#References|[a13]]] proved that $a ( G ) = t ( M _ { G } ; 2,0 )$. A variety of related evaluations appear in [[#References|[a9]]].
  
The critical problem of Crapo and G.-C. Rota [[#References|[a7]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021053.png" /> be a rank-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021054.png" /> matroid which is represented over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021055.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021056.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021057.png" /> be a positive integer. Then the number of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021058.png" />-tuples of linear functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021059.png" /> which distinguish <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021060.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021061.png" />. See [[#References|[a12]]] for extensions of the critical problem.
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The critical problem of Crapo and G.-C. Rota [[#References|[a7]]]: Let $M$ be a rank-$r$ matroid which is represented over $\operatorname {GF} ( q )$ by $\phi : E \rightarrow \operatorname {GF} ( q ) ^ { n }$ and let $k$ be a positive integer. Then the number of $k$-tuples of linear functionals on $\operatorname{GF} ( q ) ^ { n }$ which distinguish $\phi ( E )$ equals $( - 1 ) ^ { r } q ^ { k ( n - r ) } t ( M ; 1 - q ^ { k } , 0 )$. See [[#References|[a12]]] for extensions of the critical problem.
  
Coding theory: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021062.png" /> be a linear [[Code|code]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021063.png" /> with code-weight polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021065.png" /> is the weight of the code word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021066.png" /> (see also [[Coding and decoding|Coding and decoding]]). Then
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Coding theory: Let $C$ be a linear [[Code|code]] over $\operatorname {GF} ( q )$ with code-weight polynomial $A ( C ; q , z ) = \sum _ { \mathbf{v} \in C } z ^ { w (\mathbf{v}) }$, where $w ( \mathbf{v} )$ is the weight of the code word $\mathbf{v}$ (see also [[Coding and decoding|Coding and decoding]]). Then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021067.png" /></td> </tr></table>
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\begin{equation*} A ( C , q , z ) = ( 1 - z ) ^ { r } z ^ { n - r } t \left( M _ { C } ; \frac { 1 + ( q - 1 ) z } { 1 - z } , \frac { 1 } { z } \right), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021068.png" /> is the matroid associated with the code <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021069.png" />. Many standard results in coding theory have interpretations using the Tutte polynomial [[#References|[a4]]].
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where $M _ { C }$ is the matroid associated with the code $C$. Many standard results in coding theory have interpretations using the Tutte polynomial [[#References|[a4]]].
  
[[Arrangement of hyperplanes|Hyperplane arrangements]]: The number of regions in a central arrangement of hyperplanes is given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021071.png" /> is the matroid associated with the arrangement. Many generalizations of this result appear in [[#References|[a15]]].
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[[Arrangement of hyperplanes|Hyperplane arrangements]]: The number of regions in a central arrangement of hyperplanes is given by $t ( M _ { H } ; 2,0 )$, where $M _ { H }$ is the matroid associated with the arrangement. Many generalizations of this result appear in [[#References|[a15]]].
  
[[Combinatorial topology]]: The independent subsets of a matroid form a [[shellable complex complex]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021072.png" /> is the shelling polynomial associated with this complex, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021073.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021074.png" />. This result is analogous to the dichromatic interpretation for planar graphs. See [[#References|[a1]]] for more information on how the Tutte polynomial is associated to simplicial complexes.
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[[Combinatorial topology]]: The independent subsets of a matroid form a [[shellable complex]]. If $h _ { M } ( x )$ is the shelling polynomial associated with this complex, then $h _ { M } ( x ) = t ( x , 1 )$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021074.png"/>. This result is analogous to the dichromatic interpretation for planar graphs. See [[#References|[a1]]] for more information on how the Tutte polynomial is associated to simplicial complexes.
  
[[Statistical mechanics, mathematical problems in|Statistical mechanics]], network reliability and [[knot theory]]: Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021075.png" /> is a probabilistic matroid, i.e., each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021076.png" /> has an independent probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021077.png" /> of successful operation. The formula
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[[Statistical mechanics, mathematical problems in|Statistical mechanics]], network reliability and [[knot theory]]: Suppose $M$ is a probabilistic matroid, i.e., each $e \in E$ has an independent probability $p ( e )$ of successful operation. The formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021078.png" /></td> </tr></table>
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\begin{equation*} t ( M ; x , y ) = \sum _ { S \subseteq E } \left( \prod _ { e \in S } p ( e ) \right) \left( \prod _ { e \notin S } ( 1 - p ( e ) ) \right)\times  \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021079.png" /></td> </tr></table>
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\begin{equation*} \times ( x - 1 ) ^ { r ( M ) - r ( S ) } ( y - 1 ) ^ { | S |  - r ( s )} \end{equation*}
  
then produces a probabilistic Tutte polynomial. In this more general situation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021080.png" /> is the reliability of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021081.png" />, i.e., the probability that a randomly chosen subset spans <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021082.png" />. Related Tutte polynomials have applications in statistical mechanics and network reliability [[#References|[a5]]] and knot theory [[#References|[a11]]].
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then produces a probabilistic Tutte polynomial. In this more general situation, $t ( M ; 1,2 )$ is the reliability of $M$, i.e., the probability that a randomly chosen subset spans $E$. Related Tutte polynomials have applications in statistical mechanics and network reliability [[#References|[a5]]] and knot theory [[#References|[a11]]].
  
[[Greedoid]]s: When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021083.png" /> is a greedoid (cf. also [[Greedy algorithm|Greedy algorithm]]), the expansion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021084.png" /> remains valid. The recursion of T3) takes a slightly different form, however: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021085.png" /> is feasible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021086.png" />. There are many combinatorial structures which admit a greedoid rank function in a natural way, but do not possess a meaningful matroid structure. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021087.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021088.png" /> are [[rooted tree]]s (cf. also [[Tree]]), then computing the Tutte polynomial in this way gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021089.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021091.png" /> are isomorphic as rooted trees [[#References|[a8]]].
+
[[Greedoid]]s: When $G$ is a greedoid (cf. also [[Greedy algorithm|Greedy algorithm]]), the expansion $t ( G ; x , y ) = \sum_{ S \subseteq E} ( x - 1 ) ^ { r ( G ) - r ( S ) } ( y - 1 ) ^ { | S | - r ( S ) }$ remains valid. The recursion of T3) takes a slightly different form, however: If $e$ is feasible, then $t ( G ) = t ( G / e ) + ( x - 1 ) ^ { r ( G ) - r ( G - e ) } t ( G - e )$. There are many combinatorial structures which admit a greedoid rank function in a natural way, but do not possess a meaningful matroid structure. For example, if $T _ { 1 }$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021088.png"/> are [[rooted tree]]s (cf. also [[Tree]]), then computing the Tutte polynomial in this way gives $t ( T _ { 1 } ) = t ( T _ { 2 } )$ if and only if $T _ { 1 }$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021091.png"/> are isomorphic as rooted trees [[#References|[a8]]].
  
In general, it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021093.png" />-hard (cf. [[NP|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021094.png" />]]) to compute the Tutte polynomial of a graph or a matroid [[#References|[a10]]]. Certain evaluations are computable in polynomial-time for certain classes of matroids, however. For example, the number of spanning trees of a graph can be calculated in polynomial-time by the matrix-tree theorem; since this number equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t120/t120210/t12021095.png" />, this evaluation is tractable.
+
In general, it is $\cal N P$-hard (cf. [[NP|$\cal N P$]]) to compute the Tutte polynomial of a graph or a matroid [[#References|[a10]]]. Certain evaluations are computable in polynomial-time for certain classes of matroids, however. For example, the number of spanning trees of a graph can be calculated in polynomial-time by the matrix-tree theorem; since this number equals $t ( M ; 1,1 )$, this evaluation is tractable.
  
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Björner,  "The homology and shellability of matroids and geometric lattices"  N. White (ed.) , ''Matroid Applications'' , ''Encycl. Math. Appl.'' , '''40''' , Cambridge Univ. Press  (1992)  pp. 226–283</TD></TR>
+
<tr><td valign="top">[a1]</td> <td valign="top">  A. Björner,  "The homology and shellability of matroids and geometric lattices"  N. White (ed.) , ''Matroid Applications'' , ''Encycl. Math. Appl.'' , '''40''' , Cambridge Univ. Press  (1992)  pp. 226–283</td></tr>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  T.H. Brylawski,  "The Tutte–Grothendieck ring"  ''Algebra Univ.'' , '''2'''  (1972)  pp. 375–388</TD></TR>
+
<tr><td valign="top">[a2]</td> <td valign="top">  T.H. Brylawski,  "The Tutte–Grothendieck ring"  ''Algebra Univ.'' , '''2'''  (1972)  pp. 375–388</td></tr>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  T.H. Brylawski,  "The Tutte polynomial. Part 1: General theory"  A. Barlotti (ed.) , ''Matroid Theory and its Applications Proc. Third Mathematics Summer Center C.I.M.E.'' , Liguori, Naples  (1980)  pp. 125–275</TD></TR>
+
<tr><td valign="top">[a3]</td> <td valign="top">  T.H. Brylawski,  "The Tutte polynomial. Part 1: General theory"  A. Barlotti (ed.) , ''Matroid Theory and its Applications Proc. Third Mathematics Summer Center C.I.M.E.'' , Liguori, Naples  (1980)  pp. 125–275</td></tr>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  T.H. Brylawski,  J.G. Oxley,  "The Tutte polynomial and its applications"  N. White (ed.) , ''Matroid Applications'' , ''Encycl. Math. Appl.'' , '''40''' , Cambridge Univ. Press  (1992)  pp. 123–225</TD></TR>
+
<tr><td valign="top">[a4]</td> <td valign="top">  T.H. Brylawski,  J.G. Oxley,  "The Tutte polynomial and its applications"  N. White (ed.) , ''Matroid Applications'' , ''Encycl. Math. Appl.'' , '''40''' , Cambridge Univ. Press  (1992)  pp. 123–225</td></tr>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  C.J. Colbourn,  "The combinatorics of network reliability" , Oxford Univ. Press  (1987)</TD></TR>
+
<tr><td valign="top">[a5]</td> <td valign="top">  C.J. Colbourn,  "The combinatorics of network reliability" , Oxford Univ. Press  (1987)</td></tr>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  H.H. Crapo,  "The Tutte polynomial"  ''Aequat. Math.'' , '''3'''  (1969)  pp. 211–229</TD></TR>
+
<tr><td valign="top">[a6]</td> <td valign="top">  H.H. Crapo,  "The Tutte polynomial"  ''Aequat. Math.'' , '''3'''  (1969)  pp. 211–229</td></tr>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  H.H. Crapo,  G.-C. Rota,  "On the foundations of combinatorial theory: Combinatorial geometries" , MIT  (1970)  (Edition: Preliminary)</TD></TR>
+
<tr><td valign="top">[a7]</td> <td valign="top">  H.H. Crapo,  G.-C. Rota,  "On the foundations of combinatorial theory: Combinatorial geometries" , MIT  (1970)  (Edition: Preliminary)</td></tr>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  G.P. Gordon,  E.W. McMahon,  "A greedoid polynomial which distinguishes rooted arborescences"  ''Proc. Amer. Math. Soc.'' , '''107'''  (1989)  pp. 287–298</TD></TR>
+
<tr><td valign="top">[a8]</td> <td valign="top">  G.P. Gordon,  E.W. McMahon,  "A greedoid polynomial which distinguishes rooted arborescences"  ''Proc. Amer. Math. Soc.'' , '''107'''  (1989)  pp. 287–298</td></tr>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  C. Greene,  T. Zaslavsky,  "On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non–Radon partitions, and orientations of graphs"  ''Trans. Amer. Math. Soc.'' , '''280'''  (1983)  pp. 97–126</TD></TR>
+
<tr><td valign="top">[a9]</td> <td valign="top">  C. Greene,  T. Zaslavsky,  "On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non–Radon partitions, and orientations of graphs"  ''Trans. Amer. Math. Soc.'' , '''280'''  (1983)  pp. 97–126</td></tr>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  F. Jaeger,  D.L. Vertigan,  D.J.A. Welsh,  "On the computational complexity of the Jones and Tutte polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 35–53</TD></TR>
+
<tr><td valign="top">[a10]</td> <td valign="top">  F. Jaeger,  D.L. Vertigan,  D.J.A. Welsh,  "On the computational complexity of the Jones and Tutte polynomials"  ''Math. Proc. Cambridge Philos. Soc.'' , '''108'''  (1990)  pp. 35–53</td></tr>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  L. Kauffman,  "Knots and physics" , World Sci.  (1993)  (Edition: Second)</TD></TR>
+
<tr><td valign="top">[a11]</td> <td valign="top">  L. Kauffman,  "Knots and physics" , World Sci.  (1993)  (Edition: Second)</td></tr>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  J.P.S. Kung,  "Critical problems"  J. Bonin (ed.)  et al. (ed.) , ''Matroid Theory'' , ''Contemp. Math.'' , '''197''' , Amer. Math. Soc.  (1996)  pp. 1–128</TD></TR>
+
<tr><td valign="top">[a12]</td> <td valign="top">  J.P.S. Kung,  "Critical problems"  J. Bonin (ed.)  et al. (ed.) , ''Matroid Theory'' , ''Contemp. Math.'' , '''197''' , Amer. Math. Soc.  (1996)  pp. 1–128</td></tr>
<TR><TD valign="top">[a13]</TD> <TD valign="top">  R.P. Stanley,  "Acyclic orientations of graphs"  ''Discrete Math.'' , '''5'''  (1973)  pp. 171–178</TD></TR>
+
<tr><td valign="top">[a13]</td> <td valign="top">  R.P. Stanley,  "Acyclic orientations of graphs"  ''Discrete Math.'' , '''5'''  (1973)  pp. 171–178</td></tr>
<TR><TD valign="top">[a14]</TD> <TD valign="top">  W.T. Tutte,  "A contribution to the theory of chromatic polynomials"  ''Canad. J. Math.'' , '''6'''  (1954)  pp. 80–91</TD></TR>
+
<tr><td valign="top">[a14]</td> <td valign="top">  W.T. Tutte,  "A contribution to the theory of chromatic polynomials"  ''Canad. J. Math.'' , '''6'''  (1954)  pp. 80–91</td></tr>
<TR><TD valign="top">[a15]</TD> <TD valign="top">  T. Zaslavsky,  "Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes"  ''Memoirs Amer. Math. Soc.'' , '''154'''  (1975)</TD></TR>
+
<tr><td valign="top">[a15]</td> <td valign="top">  T. Zaslavsky,  "Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes"  ''Memoirs Amer. Math. Soc.'' , '''154'''  (1975)</td></tr>
<TR><TD valign="top">[a16]</TD> <TD valign="top">  T. Zaslavsky,  "The Möbius functions and the characteristic polynomial"  N. White (ed.) , ''Combinatorial Geometries'' , ''Encycl. Math. Appl.'' , '''29''' , Cambridge Univ. Press  (1987)  pp. 114–138</TD></TR>
+
<tr><td valign="top">[a16]</td> <td valign="top">  T. Zaslavsky,  "The Möbius functions and the characteristic polynomial"  N. White (ed.) , ''Combinatorial Geometries'' , ''Encycl. Math. Appl.'' , '''29''' , Cambridge Univ. Press  (1987)  pp. 114–138</td></tr>
 
</table>
 
</table>

Latest revision as of 11:16, 30 June 2021

Let $M$ be a matroid with rank function $r$ on the ground set $E$. The Tutte polynomial $t ( M ; x , y )$ of $M$ is defined by

\begin{equation*} t ( M ; x , y ) = \sum _ { S \subseteq E } ( x - 1 ) ^ { r ( M ) - r ( S ) } ( y - 1 ) ^ { | S | - r ( S )}. \end{equation*}

One also writes $t ( M )$ for $t ( M ; x , y )$ when no confusion can arise about the values of $x$ and $y$. There are several equivalent reformulations of $t ( M )$; the most useful such expression is recursive, using the matroid operations of deletion and contraction:

T0) If $E = \emptyset$, then $t ( M ) = 1$;

T1) If $e$ is an isthmus, then $t ( M ) = x t ( M / e )$;

T2) If $e$ is a loop, then $t ( M ) = y t ( M - e )$;

T3) If $e$ is neither an isthmus nor a loop, then $t ( M ) = t ( M / e ) + t ( M - e )$.

Some standard evaluations of the Tutte polynomial are:

a) $t ( M ; 1,1 )$ is the number of bases of $M$;

b) $t ( M ; 2,1 )$ is the number of independent subsets of $M$;

c) $t ( M ; 1,2 )$ is the number of spanning subsets of $M$;

d) $t ( M ; 2,2 ) = 2 ^ { | E | }$.

When a matroid $M$ is constructed from other matroids $M _ { 1 } , M _ { 2 } , \ldots$, it is frequently possible to compute $t ( M )$ from the $t ( M _ { i } )$. The most fundamental structural result of this kind concerns the direct sum of matroids: $t ( M _ { 1 } \oplus M _ { 2 } ) = t ( M _ { 1 } ) t ( M _ { 2 } )$. Another fundamental structural property of the Tutte polynomial is its transparent relationship with matroid duality: $t ( M ^ { * } ; x , y ) = t ( M ; y , x )$. Many related results can be found in [a3].

The Tutte polynomial has many significant connections with invariants in graph theory. Of central importance is the relationship between $t ( M )$ and the chromatic polynomial $\chi ( G ; \lambda )$ of a graph $G$ (cf. also Graph colouring). If $G$ is a graph having $v ( G )$ vertices and $c ( G )$ connected components, then

\begin{equation*} \chi ( G ; \lambda ) = \lambda ^ { c ( G ) } ( - 1 ) ^ { v ( G ) - c ( G ) } t ( M _ { G } , 1 - \lambda , 0 ), \end{equation*}

where $M _ { G }$ is the cycle matroid of $G$ (cf. also Matroid). Thus, when $G$ is planar (cf. Graph, planar), $t ( M _ { G } ; x , y )$ simultaneously carries information concerning proper colourings of $G$ along with proper colourings of its dual graph $G ^ { * }$. This is the reason W.T. Tutte referred to it as a dichromatic polynomial in his foundational work in this area [a14]. The polynomial was generalized from graphs to matroids by H.H. Crapo [a5] and T.H. Brylawski [a2].

Other invariants which can be obtained from the Tutte polynomial include the beta invariant $\beta ( M )$, the Möbius function $\mu ( M )$, and the characteristic polynomial $p ( M ; \lambda )$ of a matroid $M$. See [a16] for more information about these invariants.

Applications

The Tutte polynomial has applications in various areas, some of which are given below. For an extensive introduction, see [a4].

Acyclic orientations: Let $a ( G )$ denote the number of acyclic orientations of a graph $G$. Then R. Stanley [a13] proved that $a ( G ) = t ( M _ { G } ; 2,0 )$. A variety of related evaluations appear in [a9].

The critical problem of Crapo and G.-C. Rota [a7]: Let $M$ be a rank-$r$ matroid which is represented over $\operatorname {GF} ( q )$ by $\phi : E \rightarrow \operatorname {GF} ( q ) ^ { n }$ and let $k$ be a positive integer. Then the number of $k$-tuples of linear functionals on $\operatorname{GF} ( q ) ^ { n }$ which distinguish $\phi ( E )$ equals $( - 1 ) ^ { r } q ^ { k ( n - r ) } t ( M ; 1 - q ^ { k } , 0 )$. See [a12] for extensions of the critical problem.

Coding theory: Let $C$ be a linear code over $\operatorname {GF} ( q )$ with code-weight polynomial $A ( C ; q , z ) = \sum _ { \mathbf{v} \in C } z ^ { w (\mathbf{v}) }$, where $w ( \mathbf{v} )$ is the weight of the code word $\mathbf{v}$ (see also Coding and decoding). Then

\begin{equation*} A ( C , q , z ) = ( 1 - z ) ^ { r } z ^ { n - r } t \left( M _ { C } ; \frac { 1 + ( q - 1 ) z } { 1 - z } , \frac { 1 } { z } \right), \end{equation*}

where $M _ { C }$ is the matroid associated with the code $C$. Many standard results in coding theory have interpretations using the Tutte polynomial [a4].

Hyperplane arrangements: The number of regions in a central arrangement of hyperplanes is given by $t ( M _ { H } ; 2,0 )$, where $M _ { H }$ is the matroid associated with the arrangement. Many generalizations of this result appear in [a15].

Combinatorial topology: The independent subsets of a matroid form a shellable complex. If $h _ { M } ( x )$ is the shelling polynomial associated with this complex, then $h _ { M } ( x ) = t ( x , 1 )$ and . This result is analogous to the dichromatic interpretation for planar graphs. See [a1] for more information on how the Tutte polynomial is associated to simplicial complexes.

Statistical mechanics, network reliability and knot theory: Suppose $M$ is a probabilistic matroid, i.e., each $e \in E$ has an independent probability $p ( e )$ of successful operation. The formula

\begin{equation*} t ( M ; x , y ) = \sum _ { S \subseteq E } \left( \prod _ { e \in S } p ( e ) \right) \left( \prod _ { e \notin S } ( 1 - p ( e ) ) \right)\times \end{equation*}

\begin{equation*} \times ( x - 1 ) ^ { r ( M ) - r ( S ) } ( y - 1 ) ^ { | S | - r ( s )} \end{equation*}

then produces a probabilistic Tutte polynomial. In this more general situation, $t ( M ; 1,2 )$ is the reliability of $M$, i.e., the probability that a randomly chosen subset spans $E$. Related Tutte polynomials have applications in statistical mechanics and network reliability [a5] and knot theory [a11].

Greedoids: When $G$ is a greedoid (cf. also Greedy algorithm), the expansion $t ( G ; x , y ) = \sum_{ S \subseteq E} ( x - 1 ) ^ { r ( G ) - r ( S ) } ( y - 1 ) ^ { | S | - r ( S ) }$ remains valid. The recursion of T3) takes a slightly different form, however: If $e$ is feasible, then $t ( G ) = t ( G / e ) + ( x - 1 ) ^ { r ( G ) - r ( G - e ) } t ( G - e )$. There are many combinatorial structures which admit a greedoid rank function in a natural way, but do not possess a meaningful matroid structure. For example, if $T _ { 1 }$ and are rooted trees (cf. also Tree), then computing the Tutte polynomial in this way gives $t ( T _ { 1 } ) = t ( T _ { 2 } )$ if and only if $T _ { 1 }$ and are isomorphic as rooted trees [a8].

In general, it is $\cal N P$-hard (cf. $\cal N P$) to compute the Tutte polynomial of a graph or a matroid [a10]. Certain evaluations are computable in polynomial-time for certain classes of matroids, however. For example, the number of spanning trees of a graph can be calculated in polynomial-time by the matrix-tree theorem; since this number equals $t ( M ; 1,1 )$, this evaluation is tractable.

References

[a1] A. Björner, "The homology and shellability of matroids and geometric lattices" N. White (ed.) , Matroid Applications , Encycl. Math. Appl. , 40 , Cambridge Univ. Press (1992) pp. 226–283
[a2] T.H. Brylawski, "The Tutte–Grothendieck ring" Algebra Univ. , 2 (1972) pp. 375–388
[a3] T.H. Brylawski, "The Tutte polynomial. Part 1: General theory" A. Barlotti (ed.) , Matroid Theory and its Applications Proc. Third Mathematics Summer Center C.I.M.E. , Liguori, Naples (1980) pp. 125–275
[a4] T.H. Brylawski, J.G. Oxley, "The Tutte polynomial and its applications" N. White (ed.) , Matroid Applications , Encycl. Math. Appl. , 40 , Cambridge Univ. Press (1992) pp. 123–225
[a5] C.J. Colbourn, "The combinatorics of network reliability" , Oxford Univ. Press (1987)
[a6] H.H. Crapo, "The Tutte polynomial" Aequat. Math. , 3 (1969) pp. 211–229
[a7] H.H. Crapo, G.-C. Rota, "On the foundations of combinatorial theory: Combinatorial geometries" , MIT (1970) (Edition: Preliminary)
[a8] G.P. Gordon, E.W. McMahon, "A greedoid polynomial which distinguishes rooted arborescences" Proc. Amer. Math. Soc. , 107 (1989) pp. 287–298
[a9] C. Greene, T. Zaslavsky, "On the interpretation of Whitney numbers through arrangements of hyperplanes, zonotopes, non–Radon partitions, and orientations of graphs" Trans. Amer. Math. Soc. , 280 (1983) pp. 97–126
[a10] F. Jaeger, D.L. Vertigan, D.J.A. Welsh, "On the computational complexity of the Jones and Tutte polynomials" Math. Proc. Cambridge Philos. Soc. , 108 (1990) pp. 35–53
[a11] L. Kauffman, "Knots and physics" , World Sci. (1993) (Edition: Second)
[a12] J.P.S. Kung, "Critical problems" J. Bonin (ed.) et al. (ed.) , Matroid Theory , Contemp. Math. , 197 , Amer. Math. Soc. (1996) pp. 1–128
[a13] R.P. Stanley, "Acyclic orientations of graphs" Discrete Math. , 5 (1973) pp. 171–178
[a14] W.T. Tutte, "A contribution to the theory of chromatic polynomials" Canad. J. Math. , 6 (1954) pp. 80–91
[a15] T. Zaslavsky, "Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes" Memoirs Amer. Math. Soc. , 154 (1975)
[a16] T. Zaslavsky, "The Möbius functions and the characteristic polynomial" N. White (ed.) , Combinatorial Geometries , Encycl. Math. Appl. , 29 , Cambridge Univ. Press (1987) pp. 114–138
How to Cite This Entry:
Tutte polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tutte_polynomial&oldid=42759
This article was adapted from an original article by G. Gordon (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article