关于非平面图的一个广义Beraha猜想
A generalized Beraha conjecture for non-planar graphs
摘要：
研究了Q-状态Potts模型在（非平面）广义Petersen图族G（nk，k）上的配分函数Z_{G(nk,k)}（Q，v）.我们研究了它在平面（Q，v）中的零点，其中1<= k <= 7.我们还考虑了Z_{G(nk,k)}的两种特殊化，即色多项式P_{G(nk,k)}（Q）（对应于v=-1）和流多项式Phi_{G(nk,k)}（Q）（对应于v=-Q）。在这两种情况下，我们研究了它们在复Q平面中的零点，其中1 <= k <= 7。我们特别注意当n ->无穷大时相应零点的积累轨迹。我们观察到的Berker-Kadanoff阶段，是目前在二维波茨模型，也存在于非平面递归图。它们的定性特征是相同的;但主要区别是，平面图的Beraha数所起的作用现在由非平面图的非负整数所起。在Q的这些整数值处，存在大量的特征值抵消，与平面图的Beraha数处发生的特征值抵消相同。
英文摘要：
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.
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