# Graphs with the second and third maximum Wiener index over the 2-vertex connected graphs

## TL;DR

This paper characterizes the second and third maximum Wiener index graphs among 2-vertex connected graphs, extending understanding of extremal structures for this molecular descriptor.

## Contribution

It identifies and describes the specific structures of graphs with the second and third highest Wiener indices among 2-vertex connected graphs.

## Key findings

- Cycle $C_n$ has the maximum Wiener index.
- Second maximum is obtained by adding an edge between vertices at distance two on the cycle.
- Third maximum involves connecting opposite vertices of a 4-cycle with a path of length $n-3$ for large $n$.

## Abstract

Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on $n\ge 3$ vertices, the cycle $C_n$ attains the maximum value of Wiener index. We show that the second maximum graph is obtained from $C_n$ by introducing a new edge that connects two vertices at distance two on the cycle if $n\ne 6$. If $n\ge 11$, the third maximum graph is obtained from a $4$-cycle by connecting opposite vertices by a path of length $n-3$. We completely describe also the situation for $n\le 10$.