Singularities in Negami's splitting formula for the Tutte polynomial

TL;DR

This paper investigates the limitations of Negami's splitting formula for the Tutte polynomial in specific regions related to important models like the Ising and Potts models, and provides alternative splitting formulas for these cases.

Contribution

It identifies regions where Negami's splitting formula fails and derives new splitting formulas applicable to these specializations.

Findings

Negami's splitting formula is invalid on certain regions related to the Ising and Potts models.

New splitting formulas are established for these specializations.

The work connects the failure regions to important statistical models.

Abstract

The n-sum graph Negami's splitting formula for the Tutte polynomial is not valid in the region $(x-1)(y-1)=q$ for $q=1,2,\ldots n-1$ with the additional region $y=1$ if $n>3$. This region corresponds to (up to prefactors and change of variables) the Ising model, the $q$-state Potts model, the number of spanning forest generator and particularizations of these. We show splitting formulas for these specializations.