$^{GLYPH<Minus>}$30 $^{GLYPH<Minus>}$20 $^{GLYPH<Minus>}$10 10 20 30 0.01 0.02 0.03 0.04 $^{GLYPH<Minus>}$8 $^{GLYPH<Minus>}$6 $^{GLYPH<Minus>}$4 $^{GLYPH<Minus>}$2 z $^{GLYPH<Minus>}$3 $^{GLYPH<Minus>}$2 $^{GLYPH<Minus>}$1 y FIG. 4: Left: magnetisation distributions for K$_{32}$$_{,}$$_{32}$ (points) and fitted P$_{p,q}$ (64 , k ) (lines) vs k − 32 for four different temperatures. Right: y = n ( p − 1) versus z = n ( q − 1) for K$_{n/}$$_{2}$$_{,n/}$$_{2}$ , with n = 24 , 32 , 48 , 64 , 96 , 128 , 192 (downwards). Points represent K ∗ . Low temperatures in lower left corner. that the series expansion of this inverse temperature for K$_{n/}$$_{2}$$_{,n/}$$_{2}$ , i.e. a total of n vertices is 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. If the magnetisations were indeed an exact p,q -binomial distribution then we could also express the free energy as D. The free energy 33 F ( G ; K ) = K m n + log Ψ$_{p,q}$ ( n ) n (139) R ( n, n/ 2 , xn 3 / $^{4}$) ∼ exp ( − 4 3 x 4 ) (138) K ∗ = 2 n − 3 n 2 + 11 3 n 3 − 101 24 n 4 + 3827 480 n 5 + · · · (137)