Sharp bounds on the zeroth-order general Randi\'c index of trees in terms of domination number

TL;DR

This paper establishes precise upper and lower bounds for the zeroth-order general Randić index of trees based on their domination number, characterizing the extremal graphs for different parameter ranges.

Contribution

It provides the first sharp bounds on the Randić index of trees in terms of domination number for various alpha intervals, with extremal graph characterizations.

Findings

Sharp bounds on $^0R_{\alpha}$ for trees with given domination number.

Extremal graphs characterized for different alpha intervals.

Results applicable to graph theory and chemical graph analysis.

Abstract

The zeroth-order general Randi\'c index of graph $G=(V_G,E_G)$, denoted by $^0R_{\alpha}(G)$, is the sum of items $(d_{v})^{\alpha}$ over all vertices $v\in V_G$, where $\alpha$ is a pertinently chosen real number. In this paper, we obtain the sharp upper and lower bounds on $^0R_{\alpha}$ of trees with a domination number $\gamma$, in intervals $\alpha\in(-\infty,0)\cup(1,\infty)$ and $\alpha\in(0,1)$, respectively. The corresponding extremal graphs of these bounds are also characterized.