content_image/355658.jpg content_image/355659.jpg
for a graph on n vertices and m edges and it is here implied that p and q depend on K . Why this expression? Note that Z 0 = a ( m,n ) e K m = e K m and thus we have 

<img src='content_image/355658.jpg'>

from which the result follows. Compare with (121) where this relation holds exactly. Ac- tually we expect (139) to be a good approximation for K near 0, where the distributions are close to binomial, and for very high K where all the probability mass is concentrated on the extreme magnetisations. In the left plot of figure 5 we show the exactly computed free energy (red curve) for a complete bipartite graph on 16 + 16 vertices together with the p,q -approximation (139) (points). The fit is indeed very good for the whole temperature range. Taking a derivative of the points with respect to K produces a good approximation to the internal energy as the right plot shows. 

<img src='content_image/355659.jpg'>

## XI. LATTICES 

The plots in figure 4 are very representative for several graphs of interest. We intend to focus on the graphs that traditionally are studied in statistical physics; lattice graphs. We will take a look at the simple lattices in 1, 2, 3, 4 and 5 dimensions. Just to be clear, a 1D-lattice is a cycle C$_{n}$ on n vertices and it is 2-regular, i.e. 2 neighbours for each vertex. The 2-dimensional L × L -lattice is the cartesian product of two cycles on L vertices. The 

