A question arisen originally from information theory is to maximize the number of pairs of adjacent edges in a graph , given the number of vertices (n) and edges (N). The problem was first considered by Ahlswede and Katona in 1978, who characterized these extremal graphs by put forwarding the concepts of 'quasi-complete graphs' and 'quasi-stars'. They also studied the range of N (depending on n) for which one of these two extremals is superior to the other and formulated two questions on the distributions. This talk is devoted to the solution of these problems. This is a joint work with Stephan Wagner.