solid/solid transitions, transitions between graphs of different phases. (See [@AR] for a more primitive grand canonical analysis of these phases.)

We need some network notation. Consider the set $\hat G^n$ of simple graphs $G$ with set $V(G)$ of (labelled) vertices, edge set $E(G)$ and triangle set $T(G)$, where the cardinality $|V(G)|=n$. (‘Simple’ means the edges are undirected and there are no multiple edges or loops.) Think of an unordered pair of vertices as a point in an abstract space, an edge as a particle that may occupy that point, and a triangle as a many-body interaction energy associated with its edges, so the microcanonical partition function, $\displaystyle
Z^{n,\delta}_{e,t}$, is the number of simple graphs such that: $$e(G)\equiv \frac{|E(G)|}{{n \choose 2}}
\in (e-\delta,e+\delta) \quad \hbox{ and } \quad
t(G)\equiv \frac{|T(G)|}{{n\choose 3}} \in (t-\delta,t+\delta).$$ Graphs in $\displaystyle \cup_{n\ge 1}\hat G^n$ are known to have edge and triangle densities, $(e,t)$, dense in the region $R$ in the $e,t$-plane bounded by three curves, $c_1: (e,e^{3/2}), \ \ 0\le e\le
1$, the line $l_1: \ (e,0), \ \ 0\le e\le 1/2$ and a certain scalloped curve $(e,h(e)),\ \ 1/2\le e\le 1$, lying above the curve $(e,e(2e-1), \ \ 1/2\le e\le 1$, and meeting it when 