Lower bounds for the energy of weighted digraphs

ABSTRACT

Let D be a weighted digraph with weights taken from the set of non-zero real numbers and let ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$ be its eigenvalues. The energy of D is defined as $E(D)={\sum }_{k=1}^n|\mathfrak{R}{\lambda }_k|$, where $\mathfrak{R}{\lambda }_k$ denotes the real part of the complex number ${\lambda }_k$. In this paper, we obtain lower bounds for the energy of weighted digraphs in terms of weights of directed cycles of length 2. We study normal weighted digraphs and give spectral and structural characterizations of normal weighted digraphs. Using these results, we determine unicyclic and bicyclic normal weighted digraphs. For normal weighted digraphs, we obtain an improved and sharp lower bound for the energy.

KEYWORDS:

• Spectrum of a weighted digraph
• energy of a weighted digraph
• normal weighted digraph
• lower bound

AMS SUBJECT CLASSIFICATIONS:

• 05C22
• 05C50
• 05C76
• 15A18

Acknowledgements

The authors would like to thank Prof. S. Krishnan (IIT Bombay, India) for reading this paper and for useful suggestions. Finally, we thank the anonymous referee for his/her careful reading and pointing out many typos.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was partially done when first author was a post doctoral fellow at Indian Institute of Technology Bombay, India and he thanks the institute for providing fellowship [grant number AO/Admn-1/33/2015].