Maximum zeroth-order general Randi\'{c} index of orientations of trees, unicyclic and bicyclic graphs with given matching number

TL;DR

This paper determines the maximum zeroth-order general Randić index for orientations of trees, unicyclic, and bicyclic graphs based on their matching number and order, contributing to graph theory and network analysis.

Contribution

It introduces the maximum zeroth-order general Randić index for specific graph classes with given matching number, expanding understanding of graph orientation properties.

Findings

Maximum index for oriented trees, unicyclic, and bicyclic graphs identified.

Results relate the index to matching number and graph order.

Provides formulas or bounds for the index in these graph classes.

Abstract

The zeroth-order general Randi\'{c} index $R^{0}_{a}$ of a digraph $D$ is the sum of $(d^{+}_{v})^{a}+(d^{-}_{w})^{a}$ over all arcs $vw$ of $D$, where $a$, $d^{+}_{v}$ and $d^{-}_{w}$ are an arbitrary real number, the out-degree of the vertex $v$ and the in-degree of the vertex $w$, respectively. We determine maximum zeroth-order general Randi\'{c} index of oriented trees, unicyclic and bicyclic graphs in terms of matching number and order in this paper.