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

## TL;DR

This paper determines the maximum Wiener index for connected graphs with a fixed number of blocks, showing that it is achieved by a specific structure involving two cycles connected by a path.

## Contribution

It characterizes the structure of graphs with a given number of blocks that maximize the Wiener index, extending known results for special cases.

## Key findings

- Maximum Wiener index for p=1 is achieved by an n-cycle.
- Maximum Wiener index for p=n-1 is achieved by an n-path.
- For 2 ≤ p ≤ n-2, the maximum is achieved by two cycles joined by a path.

## Abstract

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).