On the vertex-degree based invariants of digraphs

TL;DR

This paper investigates extremal values of vertex-degree based invariants in digraphs, providing bounds and characterizations for various topological indices relevant in graph theory and network analysis.

Contribution

It introduces a unified approach to determine extremal values of degree-based invariants in digraphs and applies these results to several well-known topological indices.

Findings

Identified extremal digraphs for the invariant I(D)

Derived bounds for topological indices like Randić, Zagreb, and harmonic indices

Characterized extremal structures for these indices

Abstract

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\varphi$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{\varphi(d_u^+,d_v^-)}$, where $d_u^+$ (resp. $d_u^-$) denotes the out-degree (resp. in-degree) of a vertex $u$. In this paper, we give the extremal values and extremal digraphs of $I(D)$ over all digraphs with $n$ non-isolated vertices. Applying these results, we obtain the extremal values of some vertex-degree based topological indices of digraphs, such as the Randi\'{c} index, the Zagreb index, the sum-connectivity index, the $GA$ index, the $ABC$ index and the harmonic index, and the corresponding extremal digraphs.