  The analysis of large simple graphs with extreme values of the densities of
edges and triangles has been extended to the statistical structure of typical
graphs of fixed intermediate densities, by the use of large deviations of
Erdoes-Renyi graphs. We prove that the typical graph exhibits sharp
singularities as the constraining densities vary between different curves of
extreme values, and we determine the precise nature of the singularities. The
extension to graphs with fixed densities of edges and k-cycles for odd k>3 is
straightforward and we note the simple changes in the proof.
