Surface effects in dense random graphs with sharp edge constraint

We show that the random number $T_n$ of triangles in a random graph on $n$ vertices, with a strict constraint on the total number of edges, admits an expansion $T_n=a{n}^3+b{n}^2+F_n$, where $a$ and $b$ are numbers, with the mean $⟨F_n⟩=O(n)$ and the standard deviation $\sigma(T_n) =\sigma(F_n)= O(n^{3/2})$\mathrm{.}Thepresenceofa\mathrm{‘}surfaceter{m}^{\mathrm{\prime }}$bn^2$ has a significance analogous to the macroscopic surface effects of materials, and is missing in the model where the edge constraint is removed. We also find the surface effect in other graph models using similar edge constraints.