On a Variation of Randić Index

Randić index, R, also known as the connectivity or branching index, is an important topological index in chemistry. In order to attack some conjectures concerning Randić index, Dvořák et al. [5] introduced a modification of this index, denoted by R′. In this paper we present some of the basic properties of R′. We determine graphs with minimal and maximal values of R′, as well as graphs with minimal and maximal values of R′ among the trees and unicyclic graphs. We also show that if G is a triangle-free graph on n vertices with minimum degree δ, then R′(G) ≥ δ. Moreover, equality holds only for the complete bipartite graph Kδ,n−δ.