On the outdegree zeroth-order general Randić index of digraphs

Abstract

For any digraph $D$ and real number $\alpha$, the outdegree zeroth-order general Randić index is defined as the sum of the outdegree of each vertex raised to the power of $\alpha$ across all vertices in the digraph $D$. A cactus graph $G$ is a connected graph in which each block of $G$ is either an edge or a cycle. An oriented cactus is the directed variant of a cactus graph, where each edge has a specified direction. For any real number $\alpha \ge 2$ and positive integers $n$, $r$ with $n>2r$, we address the problem of identifying the maximum value of the outdegree zeroth-order general Randić index among oriented cacti with $n$ vertices and $r$ cycles. Additionally, we determine the maximum value of the outdegree zeroth-order general Randić index over connected simple digraphs with $n$ vertices and $l$ arcs, where $l$ is a positive integer. In particular, for $\alpha =2$, we obtain some of the results obtained in Ganie and Pirzada, (2024).