Zeroes of chromatic polynomials: A new approach to Beraha conjecture using quantum groups

Abstract

The number of colourings of a graphG withQ or fewer colors is a polynomial inQ known as the chromatic polynomialP _G (Q). It coïncides with the partition functionF _G of theQ state Potts model onG at zero temperature and in the antiferromagnetic regimee ^K=0. In the planar case, the Beraha conjecture particularizes the numbers$B_n = 4\cos ^2 \frac{\pi }{n}$ as possible accumulation points of real zeroes ofP _G in the infinite graph limit. We suggest in this work an approach based on recent developments of quantum groups to handle this conjecture. For the square, triangular and honeycomb lattices and systems wrapped on a cylinderl×t, we first exhibit in the (Q, e ^K) Potts parameter space a critical line, whereF _G(Q,e ^K) has real zeroes converging to and only to theB _n 's asl, t→∞. The analysis is based on the vertex representation of theQ state Potts model, quantum algebraU _qSl (2) properties forq a root of unity, and conformal invariance.U _qSl (2) symmetry is present for anye ^K, including the chromatic polynomial casee ^K=0. Using an additional hypothesis on the eigenvalues structure and knowledge of the Potts parameter space, we then argue that forP _G (Q), real zeros occur and converge toB _n 's, 2≦n≦n ₀ only, wheren ₀ depends on the lattice. Extensions to other kinds of graphs and size dependence of the zeros are discussed. Finally physical applications are also mentioned.