Singularities in the Entropy of Asymptotically Large Simple Graphs

We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. We also determine the precise modified-bipartite structure of asymptotic graphs with edge density 1/2 and triangle density in the interval $$\displaystyle [0,1/8]$$[0,1/8].