Let be an undirected graph, and let denote the cardinal number of the set of externally active edges of a spanning tree of , denote the cardinal number of the set of internally active edges of , and the number of spanning trees of whose internal activity is and external activity is . Then the Tutte polynomial, also known as the dichromate or TutteWhitney polynomial, is defined by
(1)

(Biggs 1993, p. 100).
An equivalent definition is given by
(2)

where the sum is taken over all subsets of the edge set of a graph , is the number of connected components of the subgraph on vertices induced by , is the vertex count of , and is the number of connected components of .
Several analogs of the Tutte polynomial have been considered for directed graphs, including the cover polynomial (Chung and Graham 1995), GordonTraldi polynomials (Gordon and Traldi 1993), and threevariable polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of the the GordonTraldi polynomial and 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 vertices with connected components, the Tutte polynomial is given by
(3)

where is the rank polynomial (generalizing Biggs 1993, p. 101). The Tutte polynomial is therefore a rather general twovariable graph polynomial from which a number of other important one and twovariable polynomials can be computed.
For notnecessarily connected graphs, the Tutte polynomial is related the chromatic polynomial , flow polynomial , rank polynomial , and reliability polynomial by
(4)
 
(5)
 
(6)
 
(7)

where is the number of vertices in the graph, is the number of edges, and is the number of connected components.
The Tutte polynomial of the dual graph of a graph is given by
(8)

i.e., by swapping the variables of the Tutte polynomial of the original graph. A special case of this identity relates the flow polynomial of a planar graph to the chromatic polynomial of its dual graph by
(9)

The Tutte polynomial of a connected graph is also completely defined by the following two properties (Biggs 1993, p. 103):
1. If is an edge of which is neither a loop nor an isthmus, then .
2. If is formed from a tree with edges by adding loops, then
Closed forms for some special classes of graphs are summarized in the following table, where and . The Tutte polynomial of the web graph was considered by Biggs et al. (1972) and Brennan et al. (2013).
graph  
book graph  
centipede graph  
cycle graph  
empty graph  1 
forest  
gear graph  
helm graph  
ladder graph  
ladder rung graph  
pan graph  
path graph  
star graph  
sunlet graph  
wheel graph 
The following table summarizes the recurrence relations for Tutte polynomials for some simple classes of graphs.
An equation for the Tutte polynomial of the complete graph was found by Tutte (1954, 1967). In particular, has exponential generating function
(10)

(Gessel 1995, Gessel and Sagan 1996). This can be written more simply in terms of the coboundary polynomial
(11)

where is the connected component count and is the vertex count of a graph (Martin and Reiner 2005). In this form, the exponential generating function of is given by
(12)

which can be converted to the corresponding Tutte polynomial using the above relationship and the substitution and . The formula was rediscovered by Pak in the form of the following recurrence
(13)

where .
A formula for the Tutte polynomial of a complete bipartite graph is given in terms of an bivariate exponential generating function for the coboundary polynomial as
(14)

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 unique graph and showed that wheel graphs, ladder graphs, Möbius ladders, complete multipartite graphs (with the exception of ), and hypercube graphs are unique graphs. Kuhl (2008) showed that the generalized Petersen graphs and their line graphs are unique.
The numbers of simple graphs on , 2, ... nodes that are not Tutteunique for a given value of are 0, 0, 0, 4, 15, 84, 548, 5629, ... (OEIS A243048), while the corresponding numbers of Tutteunique graphs are 1, 2, 4, 7, 19, 72, 496, 6717, ... (OEIS A243049). The following table summarizes some small coTutte graphs.
Tutte polynomial  graphs  
4  , ladder rung graph  
4  claw graph , path graph  
5  ,  
5  , ,  
5  fork graph, path graph , star graph  
5  paw graph ,  
5  bull graph, cricket graph, tadpole graph  
5  dart graph, kite graph 