it is a beautiful curve.
To deal with nodes and edges of hypercubes is my recent work. We usaully use "aggegrate" A to denote a set of connected nodes, and the size of it |A| means the number of nodes included the set. Given a size, |A| = x of an aggregate, there could be many possible different forms. Here, we want to know the number of minimum boundary edges, b(x) for all possible forms. In previous researches, Leader (1991) proposed a formula that can exactly calculate the value of b(x).
In general graph, for an aggregate, the minimum number of edges will correspond to minimum boundary nodes. However the situation is not the same.
If we want to compute the number of edges with minimum boundary nodes of an aggregate, we use function bn(x). The curve of this function will look like the graph in the upper figure. And the formula is more difficult than the previous on proposed by Leader...