A generalized Beraha conjecture for non-planar graphs Articles uri icon

publication date

  • October 2013

start page

  • 678

end page

  • 718

issue

  • 3

volume

  • 875

international standard serial number (ISSN)

  • 0550-3213

electronic international standard serial number (EISSN)

  • 1873-1562

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.

keywords

  • potts model; non-planar graphs; beraha conjecture; generalized petersen graphs; transfer matrix; berker-kadanoff phase