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 Tutte-Whitney polynomial, is defined by
(Biggs 1993, p. 100).
An equivalent definition is given by
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), Gordon-Traldi polynomials (Gordon and Traldi 1993), and three-variable -polynomial (Awan and Bernardi 2016; Chow 2016). However, with the exceptions of the the Gordon-Traldi 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 is multiplicative over disjoint unions.
For an undirected graph on vertices with connected components, the Tutte polynomial is given by
where 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.
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
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
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).
|ladder rung graph|
The following table summarizes the recurrence relations for Tutte polynomials for some simple classes of graphs.
|ladder rung graph||1|
(Gessel 1995, Gessel and Sagan 1996). This can be written more simply in terms of the coboundary polynomial
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
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
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 Tutte-unique for a given value of 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.