Some Sharp Bounds for the Spectral Radius and Energy of Digraphs

Abstract

For a simple digraph $D$ with $n$ vertices, the energy of $D$ is defined as $E(D)=\sum _{i=1}^n|\mathrm{\Re }(z_i)|$, where $z_1,z_2,\dots ,z_n$ are the eigenvalues of $D$. This paper first gives an improved lower bound on the spectral radius of $D$, which is used to obtain some upper bounds for the energy $E(D)$. These results improve and generalize some known results on upper bounds of the energy of digraphs.