For the Potts model, we build the transfer matrix by using the random-cluster represen- tation of the Potts partition function [13, 14] where the summation is over all subgraphs g of the lattice (or graph) on which the Potts model is defined, v = e K − 1, and n$_{b}$ ( g ) and n$_{c}$ ( g ) are, respectively, the number of bonds and clusters in g . For the ( q = 1) bond percolation we have simply Z = (1 + v ) $^{E}$, where E is the total number of edges of the lattice. The concept of connectivity plays an essential role in the building of the transfer matrix. Sites that belong to the same cluster are said to be connected. In the N × L cylinder, each of the L end sites of the cylinder is either isolated from or connected to other end sites. The connectivity of the L end sites of the cylinder is described by non-crossing partitions of the sites. There are a total of such non-crossing partitions [15] indexed by β . In the transfer matrix consideration the non- crossing partitions are mapped onto and coded by a set of integers 1 , 2 $^{,}$· · · , d$_{L}$ . A detailed explanation of the coding procedure can be found in [16]. The partition function of the Potts model can therefore be written as where Z ( N ) β is the partition sum restricted to the partition β . The restricted sums Z ( N ) α and Z ( N − 1) β are connected by a transfer matrix T in the form of where are elements of T . Clearly, T has the dimension $^{d$_{L}$}$×d$_{L}$ . It is also clear that the summation in (13) is over subgraphs g connecting partitions β and α of the ( N − 1)-th and N -th rows, with Δ n$_{b}$ ( g ) and Δ n$_{c}$ ( g ) denoting, respectively, the net (positive or negative) change of the number of bonds and clusters due to the introduction of g . 9 T$_{α,β}$ = ∑ g v Δ n b ( g $^{)}$qΔ n c ( g ) (13) Z ( N ) α = ∑ β T$_{α,β}$Z ( N − 1) β , (12) Z ( N ) = ∑ β Z ( N ) β , (11) d$_{L}$ = 1 2 L + 1 ( 2 L + 1 L ) (10) Z = ∑ g v n b ( g $^{)}$q n c ( g $^{)}$, (9)