On vertex peripherians and Wiener index of graphs with fixed number of cut vertices

The distance of a vertex in a graph is the sum of distances from that vertex to all other vertices of the graph. The Wiener index of a graph is the sum of distances between all its unordered pairs of vertices. A graph has been obtained that contains a vertex achieving the maximum distance among all graphs on $n$ vertices with fixed number of cut vertices. Further the graphs having maximum Wiener index among all graphs on $n$ vertices with at most $3$ cut vertices have been characterised.