Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

Abstract

We present exact results on the partition function of the q-state Potts model on various families of graphs G in a generalized external magnetic field that favors or disfavors spin values in a subset I s = {1, ..., s} of the total set of possible spin values, Z (G, q, s, v, w), where v and w are temperature-and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial P h(G, q, s, w) that counts the number of colorings of the vertices of G subject to the condition that colors of adjacent vertices are different, with a weighting w that favors or disfavors colors in the interval I s . We derive powerful new upper and lower bounds on Z (G, q, s, v, w) for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for Z (G, q, s, v, w) on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.

A note on recognizing an old friend in a new place: list coloring and the zero-temperature Potts model

“…This connection not only provides new opportunities for researchers to further reap the rewards of cross-pollination between the fields, but it also provides a formalization for, and gives an established graph theoretical foundation to, work in this area undertaken from a physics perspective. For example, it parallels the work of Shrock and Xu in [17,18], where they study a partition function that is also a specialization of the V -polynomial, but in a different limit.…”

Exact Partition Functions for the q-State Potts Model with a Generalized Magnetic Field on Lattice Strip Graphs

“…Hence, one can re-express Z(C m , q, s, v, w) in a different but equivalent form (given as Eq. (5.9) in [10]),…”

“…(2.1) and (2.2) also allow one to generalize q and s from the positive integers to the real numbers. (Indeed, studies of zeros of Z(G, q, s, v, w) in q and/or s for fixed v and w require that one generalize q and s further to the complex numbers [4,9,10].) Without loss of generality, we restrict ourselves here to connected graphs G; however, the spanning subgraphs G ′ in Eq.…”

“…As a quantity encoding information about a graph, the weighted-set chromatic polynomial P h(G, q, s, w) is more powerful than the chromatic polynomial, since it can distinguish between graphs that are chromatically equivalent, i.e., are different but have the same chromatic polynomial [4]- [10]. The zero-field q-state Potts model with partition function Z(G, q, v) is equivalent to a function of central importance in mathematical graph theory, the Tutte polynomial [11,12] (early reviews include [13,14])…”

“…Just as P h(G, q, s, w) generalizes the chromatic polynomial, so also Z(G, q, s, v, w) generalizes the Tutte polynomial. For example, Z(G, q, s, v, w) can distinguish between different graphs with the same Tutte polynomnial [9,10]. We next review some basic identities and symmetry properties of An equivalent result is obtained if one formally sets s = 0, i.e., the set of spin values that interact with the external field is the null set:…”