Maximum Randić Index on Unicyclic Graphs with \(k\) Pendant Vertices

Abstract

Let \(G\) be a graph. The Randić index of \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) denote the degrees of vertices \(u\) and \(v\) in \(G\). In this paper, we establish a sharp upper bound for the Randić index \(R(G)\) among all unicyclic graphs \(G\) with \(n\) vertices, \(k\) pendant vertices, and \(n \geq 3k\), where \(k \geq 3\).