Tutte polynomial

The Tutte polynomial, also called the dichromate or the Tutte–Whitney polynomial, is a graph polynomial. It is a polynomial in two variables which plays an important role in graph theory. It is defined for every undirected graph and contains information about how the graph is connected. It is denoted by.
The importance of this polynomial stems from the information it contains about. Though originally studied in algebraic graph theory as a generalization of counting problems related to graph coloring and nowhere-zero flow, it contains several famous other specializations from other sciences such as the Jones polynomial from knot theory and the partition functions of the Potts model from statistical physics. It is also the source of several central computational problems in theoretical computer science.
The Tutte polynomial has several equivalent definitions. It is essentially equivalent to Whitney’s rank polynomial, Tutte’s own dichromatic polynomial and Fortuin–Kasteleyn’s random cluster model under simple transformations. It is essentially a generating function for the number of edge sets of a given size and number of connected components, with immediate generalizations to matroids. It is also the most general graph invariant that can be defined by a deletion–contraction recurrence. Several textbooks about graph theory and matroid theory devote entire chapters to it.

Definitions

Definition. For an undirected graph one may define the Tutte polynomial as
where denotes the number of connected components of the graph.

In this definition it is clear that is well-defined and a polynomial in and.
The same definition can be given using slightly different notation by letting denote the rank of the graph. Then the Whitney rank generating function is defined as
The two functions are equivalent under a simple change of variables:
Tutte’s dichromatic polynomial is the result of another simple transformation:
Tutte’s original definition of is equivalent but less easily stated. For connected we set
where denotes the number of spanning trees of internal activity and external activity.
A third definition uses a deletion–contraction recurrence. The edge contraction of graph is the graph obtained by merging the vertices and and removing the edge. We write for the graph where the edge is merely removed. Then the Tutte polynomial is defined by the recurrence relation
if is neither a loop nor a bridge, with base case
if contains bridges and loops and no other edges. Especially, if contains no edges.
The random cluster model from statistical mechanics due to provides yet another equivalent definition. The partition sum
is equivalent to under the transformation

Properties

The Tutte polynomial factors into connected components. If is the union of disjoint graphs and then
If is planar and denotes its dual graph then
Especially, the chromatic polynomial of a planar graph is the flow polynomial of its dual. Tutte refers to such functions as V-functions.

Examples

Isomorphic graphs have the same Tutte polynomial, but the converse is not true. For example, the Tutte polynomial of every tree on edges is.
Tutte polynomials are often given in tabular form by listing the coefficients of in row and column. For example, the Tutte polynomial of the Petersen graph,
is given by the following table.
Other example, the Tutte polynomial of the octahedron graph is given by

History

’s interest in the deletion–contraction formula started in his undergraduate days at Trinity College, Cambridge, originally motivated by perfect rectangles and spanning trees. He often applied the formula in his research and “wondered if there were other interesting functions of graphs, invariant under isomorphism, with similar recursion formulae.” R. M. Foster had already observed that the chromatic polynomial is one such function, and Tutte began to discover more. His original terminology for graph invariants that satisfy the deletion–contraction recursion was W-function, and V-function if multiplicative over components. Tutte writes, “Playing with my W-functions I obtained a two-variable polynomial from which either the chromatic polynomial or the flow-polynomial could be obtained by setting one of the variables equal to zero, and adjusting signs.” Tutte called this function the dichromate, as he saw it as a generalization of the chromatic polynomial to two variables, but it is usually referred to as the Tutte polynomial. In Tutte’s words, “This may be unfair to Hassler Whitney who knew and used analogous coefficients without bothering to affix them to two variables.” The generalisation of the Tutte polynomial to matroids was first published by Crapo, though it appears already in Tutte’s thesis.
Independently of the work in algebraic graph theory, Potts began studying the partition function of certain models in statistical mechanics in 1952. The work by Fortuin and Kasteleyn on the random cluster model, a generalisation of the Potts model, provided a unifying expression that showed the relation to the Tutte polynomial.

Specialisations

At various points and lines of the -plane, the Tutte polynomial evaluates to quantities that have been studied in their own right in diverse fields of mathematics and physics. Part of the appeal of the Tutte polynomial comes from the unifying framework it provides for analysing these quantities.

Chromatic polynomial

At, the Tutte polynomial specialises to the chromatic polynomial,
where denotes the number of connected components of G.
For integer λ the value of chromatic polynomial equals the number of vertex colourings of G using a set of λ colours. It is clear that does not depend on the set of colours. What is less clear is that it is the evaluation at λ of a polynomial with integer coefficients. To see this, we observe:

If G has n vertices and no edges, then.
If G contains a loop, then.
If e is an edge which is not a loop, then

The three conditions above enable us to calculate, by applying a sequence of edge deletions and contractions; but they give no guarantee that a different sequence of deletions and contractions will lead to the same value. The guarantee comes from the fact that counts something, independently of the recurrence. In particular,
gives the number of acyclic orientations.

Jones polynomial

Along the hyperbola, the Tutte polynomial of a planar graph specialises to the Jones polynomial of an associated alternating knot.

Individual points

(2, 1)

counts the number of forests, i.e., the number of acyclic edge subsets.

(1, 1)

counts the number of spanning forests. If the graph is connected, counts the number of spanning trees.

(1, 2)

counts the number of spanning subgraphs.

(2, 0)

counts the number of acyclic orientations of G.

(0, 2)

counts the number of strongly connected orientations of G.

(2, 2)

is the number where is the number of edges of graph G.

(0, −2)

If G is a 4-regular graph, then
counts the number of Eulerian orientations of G. Here is the number of connected components of G.

(3, 3)

If G is the m × n grid graph, then counts the number of ways to tile a rectangle of width 4m and height 4n with T-tetrominoes.
If G is a planar graph, then equals the sum over weighted Eulerian orientations in a medial graph of G, where the weight of an orientation is 2 to the number of saddle vertices of the orientation.

Potts and Ising models

Define the hyperbola in the xy−plane:
the Tutte polynomial specialises to the partition function, of the Ising model studied in statistical physics. Specifically, along the hyperbola the two are related by the equation:
In particular,
for all complex α.
More generally, if for any positive integer q, we define the hyperbola:
then the Tutte polynomial specialises to the partition function of the q-state Potts model. Various physical quantities analysed in the framework of the Potts model translate to specific parts of the.

Potts model	Tutte polynomial
Ferromagnetic	Positive branch of
Antiferromagnetic	Negative branch of with
High temperature	Asymptote of to
Low temperature ferromagnetic	Positive branch of asymptotic to
Zero temperature antiferromagnetic	Graph q-colouring, i.e.,

Flow polynomial

At, the Tutte polynomial specialises to the flow polynomial studied in combinatorics. For a connected and undirected graph G and integer k, a nowhere-zero k-flow is an assignment of “flow” values to the edges of an arbitrary orientation of G such that the total flow entering and leaving each vertex is congruent modulo k. The flow polynomial denotes the number of nowhere-zero k-flows. This value is intimately connected with the chromatic polynomial, in fact, if G is a planar graph, the chromatic polynomial of G is equivalent to the flow polynomial of its dual graph in the sense that

Theorem.
The connection to the Tutte polynomial is given by:
Reliability polynomial
At, the Tutte polynomial specialises to the all-terminal reliability polynomial studied in network theory. For a connected graph G remove every edge with probability p; this models a network subject to random edge failures. Then the reliability polynomial is a function, a polynomial in p, that gives the probability that every pair of vertices in G remains connected after the edges fail. The connection to the Tutte polynomial is given by