On a variation of the Randíc index

The 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 the Randić index, Dvořák et al. [European J. Combin. 32 (2011), 434–442] 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 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. ∗ Also at Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, 1000 Ljubljana, Slovenia. 62 V. ANDOVA, M. KNOR, P. POTOČNIK AND R. ŠKREKOVSKI