Tutte Polynomial

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


(Biggs 1993, p. 100).

An equivalent definition is given by


where the sum is taken over all subsets A of the edge set of a graph G, k_A is the number of connected components of the subgraph on n_A vertices induced by A, n is the vertex count of G, and k is the number of connected components of G.

Several analogs of the Tutte polynomial have been considered for directed graphs, including the cover polynomial (Chung and Graham 1995), Gordon-Traldi polynomials (Gordon and Traldi 1993), and three-variable B-polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of the the Gordon-Traldi polynomial f_8 and 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).

The Tutte polynomial can be computed in the Wolfram Language using TuttePolynomial[g, {x, y}].

The Tutte polynomial is multiplicative over disjoint unions.

For an undirected graph on n vertices with c connected components, the Tutte polynomial is given by


where R(x,y) is the rank polynomial (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.

For not-necessarily connected graphs, the Tutte polynomial T(x,y) is related the chromatic polynomial pi(x), flow polynomial C^*(u), rank polynomial R(x,y), and reliability polynomial C(p) by


where n is the number of vertices in the graph, m is the number of edges, and c is the number of connected components.

The Tutte polynomial of the dual graph G^* of a graph G is given by


i.e., by swapping the variables of the Tutte polynomial of the original graph. A special case of this identity relates the flow polynomial C_G^*(u) of a planar graph G to the chromatic polynomial of its dual graph G^* by


The Tutte polynomial of a connected graph G is also completely defined by the following two properties (Biggs 1993, p. 103):

1. If e is an edge of G which is neither a loop nor an isthmus, then T_G(x,y)=T(G^((e));x,y)+T(G_((e));x,y).

2. If Lambda_(ij) is formed from a tree with i edges by adding j loops, then T(Lambda_(ij);x,y)=x^iy^j

Closed forms for some special classes of graphs are summarized in the following table, where s=sqrt((1+x+x^2+y)^2-4x^2y) and t=sqrt((x+y+1)^2-4xy). The Tutte polynomial of the web graph was considered by Biggs et al. (1972) and Brennan et al. (2013).

The following table summarizes the recurrence relations for Tutte polynomials for some simple classes of graphs.

An equation for the Tutte polynomial T_(K_n)(x,y) of the complete graph K_n was found by Tutte (1954, 1967). In particular, T_(K_n)(x,y) has exponential generating function

 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},

(Gessel 1995, Gessel and Sagan 1996). This can be written more simply in terms of the coboundary polynomial


where c_G is the connected component count and n_G is the vertex count of a graph G (Martin and Reiner 2005). In this form, the exponential generating function of K_n is given by

 1+qsum_(n=1)^inftychi^__(K_n)(q,t)(x^n)/(n!)=(sum_(n=0)^inftyt^((n; 2))(x^n)/(n!))^q,

which can be converted to the corresponding Tutte polynomial using the above relationship and the substitution q->(x-1)(y-1) and t->y. The formula was rediscovered by Pak in the form of the following recurrence

 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),

where F_n(x,y)=T_(K_(n+1))(x,y).

A formula for the Tutte polynomial of a complete bipartite graph K_(m,n) is given in terms of an bivariate exponential generating function for the coboundary polynomial as


by Martin and Reiner (2005).

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 T-unique graph and showed that wheel graphs, ladder graphs, Möbius ladders, complete multipartite graphs (with the exception of T_(1,p)), and hypercube graphs are T-unique graphs. Kuhl (2008) showed that the generalized Petersen graphs GP(m,2) and their line graphs L(GP(m,2)) are T-unique.

The numbers of simple graphs on n=1, 2, ... nodes that are not Tutte-unique for a given value of n are 0, 0, 0, 4, 15, 84, 548, 5629, ... (OEIS A243048), while the corresponding numbers of Tutte-unique graphs are 1, 2, 4, 7, 19, 72, 496, 6717, ... (OEIS A243049). The following table summarizes some small co-Tutte graphs.

nTutte polynomialgraphs
4x^2P_3 union K_1, ladder rung graph 2P_2
4x^3claw graph K_(1,3), path graph P_4
5x^2P_3 union 2K_1, 2P_2 union K_1
5x^3K_(1,3) union K_1, P_3 union P_2, P_4 union P_1
5x^4fork graph, path graph P_5, star graph S_5
5x(x+x^2+y)paw graph  union K_1, C_3 union P_2
5x^2(x+x^2+y)bull graph, cricket graph, (3,2)-tadpole graph
5x(x+2x^2+x^3+y+2xy+y^2)dart graph, kite graph

See also

Chromatic Polynomial, Flow Polynomial, Rank Polynomial, Tutte Matrix

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Tutte Polynomial

Cite this as:

Weisstein, Eric W. "Tutte Polynomial." From MathWorld--A Wolfram Web Resource.

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