Graphs with the second and third maximum Wiener indices over the 2-vertex connected graphs
Résumé
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.