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}.