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 (that is, the number of vertices with incident edges cyclicly ordered "in, out, in out"). 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 . 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 (Tutte). 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 Dichromatic polynomial Tutte also defined a closer 2-variable generalization of the chromatic polynomial, the dichromatic polynomial of a graph. This is where is the number of connected components of the spanning subgraph (V,A). This is related to the corank-nullity polynomial by The dichromatic polynomial does not generalize to matroids because k(A) is not a matroid property: different graphs with the same matroid can have different numbers of connected components. Related polynomials Martin polynomial