- NUCLEAR PHYSICS B Journal
- October 2013
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- 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.
- potts model; non-planar graphs; beraha conjecture; generalized petersen graphs; transfer matrix; berker-kadanoff phase