where m = e M − 1. To obtain an expansion in the form of (8), we define two states of the dots as either occupied or vacant. Thus there are 2 3 = 8 graphs corresponding to the 8 terms in (8). However, up to this point, clusters are defined by the connectivity of Potts sites, not by the dotted faces. But the connectivity can be readily translated to that of the dotted faces. A moment’s reflections shows that the weight contributing to A , B or C is simply q n c m n m , where n$_{c}$ is the number of independent clusters not containing sites 1, 2, or 3, and n$_{m}$ is the number of occupied dots. The rules to divide the graphs into five types corresponding to A,B,C are the same as the ones for pure 2-site interactions. For example, the graph in Fig. 7(b) has n$_{m}$ = 2, n$_{c}$ = 0 and corresponds to the term ∑ 4 , 5 , 6 ( mδ$_{1}$$_{,}$$_{5}$$_{,}$$_{6}$ )( mδ$_{2}$$_{,}$$_{4}$$_{,}$$_{6}$ ) = m $^{2}$δ$_{12}$ , thus contributing to B$_{12}$ with a term m $^{2}$. The algorithm to obtain expressions of A,B,C is therefore very similar to the one de- scribed in the above for 2-site interactions: In the Appendix we present expressions of A,B,C for the Potts model on $^{n}$× n subnets with 2-site interactions for $^{n}$≤ 4, and for subnets with 3-site interactions for $^{n}$≤ 7. We use the method of transfer matrix to calculate statistical variables for lattice models wrapped on a cylinder with circumference L and length N . For lattices shown in Fig. 1 with hatched triangles, L and N count up- and down-pointing hatched triangles (rather than individual Potts spins within each triangle). Thus, for an ( $^{m}$× m ) : ( $^{n}$× n ) lattice shown in Fig. 8(a), there are actually ( m + n ) L Potts spins in a length L . IV. THE TRANSFER MATRIX AND FINITE-SIZE SCALING 8 1. Generate one term, i.e., a graph, by choosing a set of occupied dots. 2. Count the number of clusters isolated from sites 1, 2 or 3 as n$_{c}$ . 3. Count the number of occupied dots n$_{m}$ . 4. Assign q n c m n m to A,B or C respectively according to the aforementioned rules. 5. Go to 1 for another graph until all possible graphs are exhausted.