CDAM: Computational, Discrete and Applicable Mathematics@LSE

 CDAM Research Report, LSE-CDAM-2009-01

January 2009

Tutte Polynomials of Bracelets

Norman Biggs

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 {Gn} be a family of bracelets in which the base graph has b vertices. Then the Tutte polynomial of Gn can be written as a sum of terms, one for each partition π of a non-negative integer l≤ b:
(x-1)T(Gn;x, y) = Σπ mπ(x, y) tr(Nπ(x, y))n.
The matrices N(x,y) are (essentially) the constituents of a `Potts transfer matrix', and their `multiplicities' m(x, y) are obtained by substituting k = (x-1)(y-1) in the expressions mπ(k) previously obtained in the chromatic case. As an illustration, we shall give explicit calculations for bracelets in which b is small, obtaining (for example) an exact formulae for the Tutte polynomials of the quartic plane ladders.

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