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

TL;DR

This paper investigates the extremal properties of graphs with a fixed number of cut vertices, identifying graphs with maximum Wiener index and analyzing vertex peripheries.

Contribution

It characterizes graphs with maximum Wiener index among those with at most three cut vertices and identifies vertices with maximum distance in such graphs.

Findings

Graphs with maximum Wiener index for given cut vertices are characterized.

Vertices achieving maximum distance are identified in these graphs.

Extremal graphs are explicitly constructed for specific cut vertex constraints.

Abstract

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.