The structure of graphs with given number of blocks and the maximum Wiener index

The Wiener index (the distance) of a connected graph is the sum of distances between all pairs of vertices. In this paper, we study the maximum possible value of this invariant among graphs on n vertices with fixed number of blocks p. It is known that among graphs on n vertices that have just one block, the n-cycle has the largest Wiener index. And the n-path, which has n-blocks, has the maximum Wiener index in the class of graphs on n vertices. We show that among all graphs on n vertices which have p >= 2blocks, the maximum Wiener index is attained by a graph composed of two cycles joined by a path (here we admit that one or both cycles can be replaced by a single edge, as in the case p=n - 1 for example).