Energy and Randic index of directed graphs

TL;DR

This paper extends the Randic index to directed graphs and establishes bounds relating it to the Nikiforov energy, identifying cases of equality and contributing to spectral graph theory.

Contribution

It introduces bounds connecting the Randic index and Nikiforov energy for directed graphs, with characterization of equality cases.

Findings

Proves bounds: 2R(G) ≤ E(G) ≤ 2√Δ(G) R(G)

Identifies graphs where equalities hold

Extends Randic index concept to digraphs

Abstract

The concept of Randic index has been extended recently for a digraph. We prove that $2R(G)\leq \mathcal{E}(G)\leq 2\sqrt{\Delta(G)} R(G)$, where $G$ is a digraph, and $R(G)$ denotes the Randic index, $\mathcal{E}(G)$ denotes the Nikiforov energy and $\Delta(G) $ denotes the maximum degree of $G$. In both inequalities we describe the graphs for which the equality holds.