The Randić Index and Girth of Triangle-Free Graphs

Abstract

The Randić index \(R(G)\) of a graph \(G\) is defined by \(R(G) = \sum\limits_{uv} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) is the degree of a vertex \(u\) in \(G\) and the summation extends over all edges \(uv\) of \(G\). In this work, we give sharp lower bounds of \(R(G) + g(G)\) and \(R(G) . g(G)\) among \(n\)-vertex connected triangle-free graphs with Randić index \(R\) and girth \(g\).