On a problem of Ahlswede and Katona

Let p(G) denote the number of pairs of adjacent edges in a graph G. Ahlswede and Katona considered the problem of maximizing p(G) over all simple graphs with a given number n of vertices and a given number N of edges. They showed that p(G) is either maximized by a quasi-complete graph or by a quasi-star. They also studied the range of N (depending on n) for which the quasi-complete graph is superior to the quasi-star (and vice versa) and formulated two questions on distributions in this context. This paper is devoted to the solution of these problems.