On the Zeroth-Order General Randić Index of Unicycle Graphs with \(k\) Pendant Vertices

Abstract

Let \(G\) be a graph. The zeroth-order general Randić index of a graph is defined as \(R_\alpha^0(G) = \sum_{v \in V(G)} d(v)^\alpha(v)\), where \(\alpha\) is an arbitrary real number and \(d(v)\) is the degree of the vertex \(v\) in \(G\). In this paper, we give sharp lower and upper bounds for the zeroth-order general Randić index \(R_\alpha^0(G)\) among all unicycle graphs \(G\) with \(n\) vertices and \(k\) pendant vertices.