On the spectral radius and energy of digraphs

Abstract

Let D be a simple digraph with eigenvalues z₁, z₂, …, z_n. The energy of D is defined as $E\left(D\right)=\sum _{i=1}^n|\mathrm{R}\mathrm{e}\left(z_i\right)|$, where Re(z_i) is the real part of the eigenvalue z_i. In this paper, a lower bound for the spectral radius of D will be established based on the number of subgraphs $\overleftrightarrow{P_3}$ in D, improving some of the lower bounds that appear in the literature. Furthermore, this result allows us to obtain an upper bound for the energy of D.

Keywords:

• Digraph
• energy
• spectral radius

2010 Mathematics Subject Classifications:

• 15A18
• 05C20

Acknowledgements

We would like to thank the anonymous referees for the careful reading of our manuscript and for the invaluable comments. J. Rodríguez was supported by MINEDUC-UA project, code ANT-1899 and Funded by the Initiation Program in Research – Universidad de Antofagasta, INI-19-06.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

J. Rodríguez was supported by MINEDUC-UA project, code ANT-1899 and Funded by the Initiation Program in Research – Universidad de Antofagasta, INI-19-06.