where a ′ is an unknown constant. Because y$_{u}$ < 0 and y$_{t}$ > 0, v$_{c}$ ( L ) for a sequence of increasing system sizes converge to the critical point v$_{c}$ . At v$_{c}$ ( L ), the expression X$_{h}$ ( v$_{c}$ ( L ) , L ) for a sequence of sizes L converge to the magnetic scaling dimension X$_{h}$ as with b ′ an unknown constant. This determines the magnetic scaling dimension X$_{h}$ . The free energy per unit distance is given by where λ$_{0}$ is the largest eigenvalue of the transfer matrix in the non-magnetic sector. Accord- ing to conformal invariance theory, the large- L asymptotic finite-size behavior of the free energy density at the critical point is [27, 28] where c is the conformal anomaly. The conformal anomaly c and the magnetic scaling dimension X$_{h}$ are two important universal quantities defining the universality class. For the two-dimensional q -state Potts model, they are given by the conformal invariance theory and Coulomb gas method [3, 4, 27– 29] as where In this section we present numerical results of our transfer matrix calculations and finite- size scaling analysis for the 3-12 and kagome-type lattices. We also present some exact $^{results for site percolation on the (1}$× 1) : ( $^{n}$× n ) lattices. Critical points are estimated by extrapolating the solutions of (25) for a sequence of in- creasing system sizes in accordance with the finite-size scaling equation (26). The numerical 14 A. The q -state Potts model on the 3-12 and kagome-type subnet lattices V. NUMERICAL AND SOME EXACT RESULTS X$_{h}$ ( v$_{c}$ ( L ) , L ) = X$_{h}$ + b $^{′}$uL y u + ... , (27) f ( L ) = ζ ln λ$_{0}$ L , (28) f ( L ) ≃ f $^{(}$∞) + πc 6 L 2 , (29) c = 1 − 6(1 − g ) 2 g , X$_{h}$ = 1 − g 2 − 3 8 g , (30) √ q = − 2 cos( πg ) , 1 2 ≤ g ≤ 1 . (31)