A generalized Beraha conjecture for non-planar graphs

TL;DR

This paper investigates the zeros of the Potts model partition function on non-planar generalized Petersen graphs, revealing that the Beraha number phenomena extend to non-planar graphs with integers replacing Beraha numbers.

Contribution

It extends the understanding of the Beraha conjecture to non-planar graphs, showing similar eigenvalue cancellation phenomena at integers instead of Beraha numbers.

Findings

Berker-Kadanoff phase exists in non-planar graphs

Eigenvalue cancellations occur at non-negative integers

Qualitative features of zeros are similar to planar graphs

Abstract

We study the partition function Z_{G(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 Z_{G(nk,k)}, namely the chromatic polynomial P_{G(nk,k)}(Q) (corresponding to v=-1), and the flow polynomial Phi_{G(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 -> infinity. 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.