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 $$n-1$$ n - 1 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 $$p\ge 2$$ p ≥ 2 blocks, 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 $$p=n-1$$ p = n - 1 for example).