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 {G_n} be a family of bracelets in which the base graph has b vertices. Then the Tutte polynomial of G_n can be written as a sum of terms, one for each partition π of a non-negative integer l≤ b: (x-1)T(G_n;x, y) = Σ_π m_π(x, y) tr(N_π(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 m_π(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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		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk