Energy and Randić index of directed graphs

Abstract

The concept of Randić index has been extended recently for a digraph. We prove that $2R(G)\le \mathcal{E}(G)\le 2\sqrt{\mathrm{\Delta }(G)}R(G)$, where G is a digraph, and $R(G)$ denotes the Randić index, $\mathcal{E}(G)$ denotes the Nikiforov energy and $\mathrm{\Delta }(G)$ denotes the maximum degree of G. Moreover, in both inequalities, we describe the graphs for which the equality holds. We also develop the concept of inner and outer energies of a vertex for directed graphs extending important inequalities known for the energy of a vertex for graphs.

COMMUNICATED BY:

• J. Y. Shao

Keywords:

• Randić index
• graph energy
• digraphs
• vertex energy

MSC 2010:

• 05C50
• 05C09

Acknowledgments

The author thanks Roland Speicher and Moritz Weber for the nice environment at University of Saarlands, while visiting for a sabbatical period.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

O.A. received support from Conacyt (Consejo Nacional de Ciencia y Tecnología) [grant number CB-2017-2018-A1-S-9764] and from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no 734922.