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 >= 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 not equal 6. If n >= 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 <= 10.