the sum taken over all vertices i of G . Note that − n ≤ M ≤ n and it only takes every alternate value, i.e. M ∈ {− n, − n + 2 , − n + 4 , . . ., n − 4 , n − 2 , n } . We will often need to refer to it in terms of how many negative spins the state has. If k spins are negative then M = n − 2 k . The partition function of the Ising model is defined for any graph G as The coefficients a ( E, M ) then are defined as the number of states with energy E and mag- netisation M . Denote the number of states at energy E by a ( E ) = ∑ M a ( E, M ). Note that the number of states at magnetisation M is just ( n k ) , where k = ( n − M ) / 2 is the number of negative spins. Let also Z$_{k}$ denote the terms of Z with magnetisation M = n − 2 k for a graph on n vertices, so that Z = Z$_{0}$ + Z$_{2}$ + · · · + Z$_{n}$, i.e. Z$_{k}$ are the terms corresponding to k negative spins. If we evaluate the partition function in x = e K and y = e H with K the dimension- less coupling, or inverse temperature J/k$_{B}$T , and H = h/k$_{B}$T as the dimensionless exter- nal magnetic field, we obtain the physical partition function denoted Z = Z ( G ; K, H ) = Z( G ; e $^{K}$, e$^{H}$), though we are usually interested only in the case when H = 0 (or y = 1). Analogously, we write Z = Z 0 + · · · + Z $_{n}$. The dimensionless and normalised free energy is defined as F = (log Z ) /n . From the derivatives of the free energy we can now obtain other physical quantities such as the internal energy ∂ F /∂K and the specific heat ∂ $^{2}$F /∂K 2 though we shall not be needing the latter for this investigation. We assume the Boltzmann distribution on the states so that (with H = 0) the probability for state τ is We have then especially that the probability for energy E is and the probability for magnetisation M is where M = n − 2 k . 29 P ( τ ) = e K E ( τ ) Z (115) P ( E ) = a ( E ) e K E Z (116) P ( M ) = 1 Z ∑ E a ( E, M ) e K E = Z k (117) Z( G ; x, y ) = ∑ τ x E ( τ $^{)}$y M ( τ ) = ∑ E,M a ( E, M ) x $^{E}$yM (114)