Tutte Polynomials of Bracelets
نویسنده
چکیده
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 ` ≤ b: (x− 1)T (Gn;x, y) = ∑ π mπ(x, y) tr(Nπ(x, y)) . The matrices Nπ(x, y) are (essentially) the constituents of a ‘Potts transfer matrix’, and their ‘multiplicities’ mπ(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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