A generalized Beraha conjecture for non-planar graphs

We study the partition function image of the Q-state Potts model on the family of (non-planar) generalized Petersen graphs image. We study its zeros in the plane image for image. We also consider two specializations of image, namely the chromatic polynomial image (corresponding to image), and the flow polynomial image (corresponding to image). In these two cases, we study their zeros in the complex Q-plane for image. We pay special attention to the accumulation loci of the corresponding zeros when image. 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.