Tutte polynomials of bracelets

Biggs, Norman

Journal of Algebraic Combinatorics, Tome 32 (2010) no. 3, pp. 389-398.

Résumé

Summary: 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 $\ell \leq b: ( x -1) T( G _{ n}; x, y)$= å $_{ p} m _{ p}( x, y)\operatorname tr( N _{ p}( x, y)) ^{ n}$. (x-1)$T(G_n;x,y)=$sum_pim_pi(x,y)$\operatorname $trbigl(N_pi(x,y)bigr)^n. 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.

Détail

Keywords: keywords tutte polynomial, Potts model, transfer matrix, Specht modules