Maximum zeroth-order general Randi\'{c} index of orientations of cacti

TL;DR

This paper identifies orientations of cactus graphs that maximize the zeroth-order general Randić index for real parameter a ≥ 1, contributing to the understanding of degree-based graph invariants.

Contribution

It determines the orientations of cacti that achieve the maximum zeroth-order general Randić index for a ≥ 1, a novel optimization in graph orientation theory.

Findings

Identified maximum Randić index orientations for cacti.

Provided explicit conditions for optimal orientations.

Extended understanding of degree-based graph invariants.

Abstract

The zeroth-order general Randi\'{c} index $R^{0}_{a+1}$ of an $n$-vertices oriented graph $D$ is equal to the sum of $(d^{+}_{u_i})^{a}+(d^{-}_{u_j})^{a}$ over all arcs $u_iu_j$ of $D$, where we denote by $d^{+}_{u_i}$ the out-degree of the vertex $u_i$ and $d^{-}_{u_j}$ the in-degree of the vertex $u_j$, $a$ is an arbitrary real number. In the paper, we determine the orientations of cacti with the maximum value of the zeroth-order general Randi\'{c} index for $a\geq 1$.