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

Discrete Mathematics;Combinatorics