content_image/347958.jpg content_image/347968.jpg content_image/347967.jpg content_image/347966.jpg
<img src='content_image/347958.jpg'>

that the series expansion of this inverse temperature for K$_{n/}$$_{2}$$_{,n/}$$_{2}$ , i.e. a total of n vertices is 

<img src='content_image/347968.jpg'>

The calculations behind this are rather long and were done with Mathematica. Compare this with the expansion in the previous subsection for the complete graph on n vertices, K$_{n}$ , which begins K ∗ = 1 /n − 1 /n 2 + 4 / 3 n 3 + · · · . In the lower left corner the distribution has P ( M = n ) = P ( M = n − 2), i.e. at K = log n/n . The points in the right panel of figure 4 should approach z = y = − 2. One can show that the distribution of a balanced bipartite graph K$_{n/}$$_{2}$$_{,n/}$$_{2}$ has the shape 

at K ∗ , just like the complete graph on n vertices. Since Φ( − 2) = 4 / 3 we then assume that z , and thus also y , will approach − 2. 

## D. The free energy 

<img src='content_image/347967.jpg'>

If the magnetisations were indeed an exact p,q -binomial distribution then we could also express the free energy as 

<img src='content_image/347966.jpg'>

