A generalized Beraha conjecture for non-planar graphs

Abstract

We study the partition function ZG(nk,k) (Q,v) of the Q -state Potts model on the family of (non-planar) generalized Petersen graphs G(nk,k). We study its zeros in the plane (Q,v) for 1≤k≤7. We also consider two specializations of ZG(nk,k), namely the chromatic polynomial PG(nk,k) (Q) (corresponding to v= –1), and the flow polynomial PhiG(nk,k) (Q) (corresponding to v= –Q). In these two cases, we study their zeros in the complex Q -plane for 1≤k≤7. We pay special attention to the accumulation loci of the corresponding zeros when n→∞. We observe that the Berker&-Kadanoff phase that is present in two-dimensional Potts models, also exists for non-planar recursive graphs. Their qualitative features are the same; but the main difference is that the role played by the Beraha numbers for planar graphs is now played by the non-negative integers for non-planar graphs. At these integer values of Q, there are massive eigenvalue cancellations, in the same way as the eigenvalue cancellations that happen at the Beraha numbers for planar graphs.