On \(R’\) Index of a Graph

Abstract

The Randić index \(R\) is an important topological index in chemistry. In order to attack some conjectures concerning the Randić index, a modification \(R’\) of this index was introduced by Dvorak et al. [6]. The \(R’\) index of a graph \(G\) is defined as the sum of the weights \(\frac{1}{\max\{{d(u)d(v)}\}}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). We first give a best possible lower bound of \(R’\) for a graph with minimum degree at least two and characterize the corresponding extremal graphs, and then we establish some relations between \(R’\) and the chromatic number, the girth of a graph.