The Connectivity and the Harary Index of a Graph

TL;DR

This paper investigates the Harary index of graphs, providing bounds based on connectivity measures and characterizing extremal graphs with maximum or second maximum Harary index for given connectivity constraints.

Contribution

It introduces bounds on the Harary index related to graph connectivity and characterizes extremal graphs achieving these bounds.

Findings

Identified upper bounds for the Harary index based on connectivity.

Characterized the unique graphs with maximum Harary index for given connectivity.

Determined extremal graphs with second maximum Harary index.

Abstract

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We characterize the unique graph with maximum Harary index among all graphs with given number of cut vertices or vertex connectivity or edge connectivity. In addition we also characterize the extremal graphs with the second maximum Harary index among the graphs with given vertex connectivity.