content_image/360282.jpg content_image/360281.jpg
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 K ∗ = 1 log 1 + 2 for K$_{n}$ . 

2 ( 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. 

## B. The average graph 

Let us compute the sum of all partition functions taken over all graphs on n vertices. 

<img src='content_image/360282.jpg'>

and for y = 1 we have 

<img src='content_image/360281.jpg'>

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. 

