Entire Harary index of graphs

Abstract

The Harary index H(G) of a connected graph G is defined as the sum of reciprocals of distances between all pairs of vertices in G [9, 13]. In this paper, as a continuing the study of entire indices like entire Zagreb indices of graphs [1], we introduce the entire Harary index by adding the reciprocals of distances between all pairs of edges and all pairs of edges and vertices to the Harary index. Our motivation in doing this coming from the following fact about molecular graphs: The intermolecular forces do not exist only between the atoms, but also between the atoms and bonds, so the relations between the atoms and bonds and also between the bonds together should be taken in account. Exact values of some families of graphs and some graphs operations are obtained. Simple formula of the entire Harary index of trees and some bounds are established.

Subject Classification:

• 05C05
• 05C07
• 05C90

Keywords:

• Graph
• Harary index
• Entire Harary index