page_title: Tutte polynomial
section_title: Potts and Ising models
hierarchical_section_title: Tutte polynomial / Specialisations / Potts and Ising models
caption_attribution_description: English: The partition function of the Potts and Ising models drawn in the Tutte plane
context_page_description: 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 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 connected components, with immediate generalizations to matroids.
context_section_description: 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 .
caption: The partition functions for the Ising model and the 3- and 4-state Potts models drawn in the Tutte plane.
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