K ∗ = 2 log ( 1 + n ) for K$_{n}$ . In short, the partition function and the magnetisation distribution for K$_{n}$ can be ex- pressed in terms of p, q -binomial coefficients. Does this hold for all graphs? No. In fact, it seems to only be true for K$_{n}$ . However, it does seem to hold asymptotically as the order of the graphs increase, for some interesting classes of graphs. The precise formulation of such a statement remains and falls outside this paper. Since we have defined the critical temperature as the K = K ∗ where the middle ratio is 1, i.e. P ( M = − 2) = P ( M = 0) = P ( M = +2) then this corresponds to the point where R$_{q,q}$ ( n, n/ 2 , 1) = 1, which takes place at q = n/ ( n + 2) as we saw in lemma IV.1. Thus 1 2 Let us compute the sum of all partition functions taken over all graphs on n vertices. B. The average graph and for y = 1 we have where q = 1 /x so that K ∗ = log(1+2 /n ). Again we have P ( M = n − 2 k ) = P$_{q,q}$ ( n, k ). The mean magnetisation distribution can then be modelled by p, q -binomial coefficients, though with p = q , just as for the complete graph. 31 ¯ Z$_{n}$( x, 1) = (1 + x ) ( n $_{2}$) n ∑ i =0 ( n i ) ( 1 x ) i ( $^{n}$−i ) = (127) (1 + x ) ( n $_{2}$) n ∑ i =0 [ n i ] q,q = (1 + x ) ( n $_{2}$) Ψ$_{q,q}$ ( n ) (128) ¯ Z$_{n}$( x, y ) = ∑ $^{G}$⊆K$_{n}$ Z( G ; x, y ) = (123) n ∑ i =0 ( n i ) y $^{n}$− 2 i ( i $_{2}$) ∑ j =0 (( i 2 ) j ) x j ( n − i 2 ) ∑ k =0 (( $^{n}$−i 2 ) k ) x k i ( $^{n}$−i ) ∑ ℓ =0 ( i ( n − i ) ℓ ) x − ℓ = (124) n ∑ i =0 ( n i ) y $^{n}$− 2 i (1 + x ) ( i $_{2}$) (1 + x ) ( n − i 2 ) ( 1 + 1 x ) ( i ( n − i ) 2 ) = (125) (1 + x ) ( n $_{2}$) y n n ∑ i =0 ( n i ) ( 1 y 2 ) i ( 1 x ) i ( $^{n}$−i ) (126)