2 FIG. 1: Topology of (3 , 4 , 6 , 4) [left] and (3 $^{4}$, 6) [right] AL. II. MODEL AND SIMULATION We consider the two-dimensional Ising S = 1 model on (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL lattices. The Hamiltonian of the system can be written as where spin variable S$_{i}$ takes values − 1, 0, +1 and deco- rates every N = 6 L 2 vertex of the AL. In Eq. (1) J is the magnetic exchange coupling parameter. The simulations have been performed for different lat- tice sizes L = 8, 16, 32, 64 and 128. For each system with N = 6 L 2 spins and given temperature T we per- formed Monte Carlo simulation in order to evaluate the system magnetisation m . The simulations start with a uniform configuration of spins ( S$_{i}$ = +1, but the results are independent on the initial configuration). It takes 10 5 Monte Carlo steps (MCS) per spin for reaching the steady state, and then the time average over the next 10 5 MCS are estimated. One MCS is accomplished when all N spins are investigated whether they should flip or not. We carried out N$_{run}$ = 20 to 50 independent simulations for each lattice and for given set of parameters ( N, T ). We have employed the heat bath algorithm for the spin dynamic. We evaluate the average magnetisation M , the suscep- tibility χ , and the magnetic 4-th order cumulant U : where m = ∑ $_{i}$S$_{i}$ /N and k$_{B}$ is the Boltzmann constant. In the above equations 〈 ... 〉 stands for thermodynamic average. In the infinite-volume limit these quantities (2) exhibit singularities at the transition point T$_{C}$ . In finite systems the singularities are smeared out and scale in the critical region according to where ν , β and γ are the usual critical exponents, and f$_{i}$ ( x ) are finite size scaling (FSS) functions with x = ( T − T$_{C}$ ) L 1 /ν being the scaling variable. Therefore, from the size dependence of M and χ one can obtain the exponents β/ν and γ/ν , respectively. The maximum value of susceptibility also scales as L $^{γ/ν}$. Moreover, the value of temperature T ∗ for which χ has a maximum, is expected to scale with the system size as where the constant b is close to unity [23]. Therefore, the Eq. (4) may be used to determine the exponent 1 /ν . We have checked also if the calculated exponents satisfy the hyper-scaling hypothesis in order to get the effective dimensionality, D$_{eff}$ , for both investigated AL lattices. The dependence of the magnetisation M on the tem- perature T , obtained from simulations on (3 , 4 , 6 , 4) and (3 $^{4}$, 6) AL with N = 6 L 2 ranging from 384 to 98304 sites is presented in Fig. 2. The shape of magnetisation curve versus temperature, for a given value of N , suggests the presents of the second-order transition phase in the sys- tem. The phase transition occurs at the critical value T$_{C}$ of temperature. In order to estimate the critical temperature T$_{C}$ we calculate the fourth-order Binder cumulants given by Eq. (2c). It is well known that these quantities are indepen- dent of the system size at T$_{C}$ and should intercept there [24]. III. RESULTS AND DISCUSSION M = L − $^{β/ν}$f$_{M}$ ( x ) , (3a) χ = L − $^{γ/ν}$f$_{χ}$ ( x ) , (3b) T $^{∗}$( L ) = T$_{C}$ + bL − 1 $^{/ν}$, (4) 2 β/ν + γ/ν = D$_{eff}$ (5) H = − J N ∑ i =1 N ∑ j>i S$_{i}$S$_{j}$, (1) M ( T, L ) = 〈| m |〉 , (2a) k$_{B}$T J · χ ( T, L ) = N ( 〈 m $^{2}$〉 − 〈 | m |〉 $^{2}$) , (2b) U ( T, L ) = 1 − 〈 m $^{4}$〉 3 〈| m |〉 $_{2}$, (2c)