Some Bounds on Zeroth-Order General Randi\'c$ Index

TL;DR

This paper extends bounds on the zeroth-order general Randić index for graphs, exploring cases where the parameter is negative, and characterizes extremal graphs for these bounds.

Contribution

It generalizes previous bounds on the inverse degree to the case of negative gamma and characterizes the extremal graphs in this setting.

Findings

Extended bounds on the zeroth-order general Randić index for gamma<0.

Characterized extremal graphs achieving these bounds.

Connected graphs' properties related to the index are analyzed.

Abstract

For a graph $G$ without isolated vertices, the inverse degree of a graph $G$ is defined as $ID(G)=\sum_{u\in V(G)}d(u)^{-1}$ where $d(u)$ is the number of vertices adjacent to the vertex $u$ in $G$. By replacing $-1$ by any non-zero real number we obtain zeroth-order general Randi\'c index, i.e. $^0R_{\gamma}(G)=\sum_{u\in V(G)}d(u)^{\gamma}$ where $\gamma$ is any non-zero real number. In \cite{xd}, Xu et. al. determined some upper and lower bounds on the inverse degree for a connected graph $G$ in terms of chromatic number, clique number, connectivity, number of cut edges. In this paper, we extend their results and investigate if the same results hold for $\gamma<0$. The corresponding extremal graphs have been also characterized.