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 , and 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 [18] 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 - 2 s - c ( - 1 < c < 1), which extends partial results above.