Denote by K ∗ the coupling where Z n/ $^{2}$−1 = Z n/ 2 = Z n/ $_{2+1}$. This coupling will corre- spond to the relation R$_{p,q}$ ( n, n/ 2 , 1) = 1 for some choice of p, q . We define the (sponta- neous) normalised magnetisation ¯ µ = 〈| M |〉 /n and the (spontaneous) susceptibility ¯ χ = Var ( | M | ) /n = ( 〈 M $^{2}$〉 − 〈 | M |〉 $^{2}$) /n . The pure susceptibility is simply χ = Var ( M ) /n = 〈 M $^{2}$〉 = 4 σ$_{2}$/n . Since M = n − 2 k we thus have ¯ µ = 2 σ$_{1}$/n and ¯ χ = 4 ( σ$_{2}$ − σ 2 $_{1}$) /n . Recall the traditional finite-size scaling laws which claim that in the critical region, i.e. near K$_{c}$ , ¯ µ ∝ L − β/ν and ¯ χ ∝ L $^{γ/ν}$. Being near K$_{c}$ means that | K − K$_{c}$ | ∝ L − 1 /ν and we especially expect K ∗ to belong to this region. Though the high- and low-temperature exponents may or may not be equal for three dimensions, see 12 for an in-depth numerical investigation of this matter, the details of these exponents are not important for our present investigation. What matters is that there are exponents that guide the growth of e.g. the susceptibility near K$_{c}$ . For a complete graph, denoted K$_{n}$ on n vertices and ( n 2 ) edges the partition function is easy to compute. Suppose k of the vertices are assigned spin − 1 and the other n − k have spin +1. The magnetisation is obviously M = n − 2 k and the energy is The partition function is then and with y = 1 we have where q = 1 /x $^{2}$. Thus we have where q = exp( − 2 K ). Obviously we have A. The complete graph E = ( k 2 ) + ( n − k 2 ) − k ( n − k ) = ( n 2 ) − 2 k ( n − k ) (118) Z( K$_{n}$ ; x, y ) = x ( n $_{2}$) y n n ∑ k =0 ( n k ) ( 1 x 2 ) k ( $^{n}$−k ) ( 1 y 2 ) k (119) Z( K$_{n}$ ; x, 1) = x ( n $_{2}$) n ∑ k =0 [ n k ] q,q = x ( n $_{2}$) Ψ$_{q,q}$ ( n ) (120) Z ( K$_{n}$ ; K, 0) = exp { K ( n 2 )} Ψ$_{q,q}$ ( n ) (121) P ( M = n − 2 k ) = P$_{q,q}$ ( n, k ) (122) 30