Complexity: knots, colourings and counting
Complexity: knots, colourings and counting
The Brown--Colbourn conjecture on zeros of reliability polynomials is false
Journal of Combinatorial Theory Series B
Specht modules and chromatic polynomials
Journal of Combinatorial Theory Series B - Special issue dedicated to professor W. T. Tutte
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The identity linking the Tutte polynomial with the Potts model on a graph implies the existence of a decomposition resembling that previously obtained for the chromatic polynomial. Specifically, let {G n } be a family of bracelets in which the base graph has b vertices. It is shown here (Theorems 3 and 4) that the Tutte polynomial of G n can be written as a sum of terms, one for each partition 驴 of a nonnegative integer 驴驴b: $$(x-1)T(G_n;x,y)=\sum_{\pi}m_{\pi}(x,y)\operatorname {tr}\bigl(N_{\pi}(x,y)\bigr)^n.$$ The matrices N 驴 (x,y) are (essentially) the constituents of a `Potts transfer matrix', and a formula for their sizes is obtained. The multiplicities m 驴 (x,y) are obtained by substituting k=(x驴1)(y驴1) in the expressions m 驴 (k) previously obtained in the chromatic case. As an illustration, explicit calculations are given for some small bracelets.