Graphs with a given conditional diameter that maximize the Wiener index

The Wiener index W (G) of a graph G is one of the most well-known topological indices, which is defined as the sum of distances between all pairs of vertices of G. The diameter D (G) of G is the maximum distance between all pairs of vertices of G; the conditional diameter D (G;s) is the maximum distance between all pairs of vertex subsets with cardinality s of G. When s=1, the conditional diameter D (G;s) is just the diameter D (G). The authors in QS characterized the graphs with the maximum Wiener index among all graphs with diameter D (G) =n-c, where 1 c 4. In this paper, we will characterize the graphs with the maximum Wiener index among all graphs with conditional diameter D (G;s) =n-2s-c (-1 c 1), which extends partial results in QS.