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Tutte polynomials of bracelets

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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 pi of a nonnegative integer a""a parts per thousand currency signb: The matrices N (pi) (x,y) are (essentially) the constituents of a 'Potts transfer matrix', and a formula for their sizes is obtained. The multiplicities m (pi) (x,y) are obtained by substituting k=(x-1)(y-1) in the expressions m (pi) (k) previously obtained in the chromatic case. As an illustration, explicit calculations are given for some small bracelets.

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