On the energy of digraphs

TL;DR

This paper establishes new bounds on the spectral radius and energy of directed graphs, improving previous results and characterizing cases where these bounds are tight.

Contribution

It introduces a new lower bound for the spectral radius of digraphs, leading to an improved upper bound for their energy, with characterizations of extremal cases.

Findings

New lower bound for spectral radius of digraphs

Improved upper bound for digraph energy

Characterization of digraphs where bounds are tight

Abstract

Let $D$ be a simple digraph with eigenvalues $z_1,z_2,...,z_n$. The energy of $D$ is defined as $E(D)= \sum_{i=1}^n |Re(z_i)|$, is the real part of the eigenvalue $z_i$. In this paper a lower bound will be obtained for the spectral radius of $D$, wich improves some the lower bounds that appear in the literature \cite{G-R}, \cite{T-C}. This result allows us to obtain an upper bound for the energy of $ D $. Finally, digraphs are characterized in which this upper bound improves the bounds given in \cite{G-R} and \cite{T-C}.