On spectra and real energy of complex weighted digraphs

ABSTRACT

In this paper, we study spectral properties of complex weighted digraphs. We show that a complex weighted digraph D is balanced if and only if D and $|D|$ have the same spectrum, where $|D|$ is the absolute value weighted digraph of D, that is, the digraph obtained by replacing the weight of each arc by its absolute value. We extend the concept of real energy to complex weighted digraphs and obtain extremal energy unicyclic complex weighted digraphs with cycle-weight in the punctured disk $\mathrm{\Delta }=\{z\in \mathbb{C}:|z|\le 1\}\mathrm{\setminus }\left\{0\right\}$. We consider a family of complex weighted digraphs ${\mathcal{D}}_{n,h}$, in which each digraph has order n and cycles of length $h\ge 2$ only with constant complex weight $c=a+\dot{\iota }b$. We show that for each $D\in {\mathcal{D}}_{n,h}$, the real energy of D is related to the real energy of unweighted cycle of length h and in some special cases real energy can be compared using quasi-order relation on coefficients of the characteristic polynomial. Finally, we obtain upper bounds on the real energy which generalize those known for unweighted digraphs and signed digraphs.

COMMUNICATED BY:

• Ravindra B. Bapat

Keywords:

• Spectrum of a complex weighted digraph
• balance
• real energy of a complex weighted digraph
• bounds on the real energy of a complex weighted digraph

AMS Subject Classifications:

• 05C22
• 05C50
• 05C76
• 15A18

Acknowledgments

The authors thank the anonymous referees for their careful reading and useful suggestions which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author is thankful to SERB-DST, New Delhi, India, for providing support under research project number EMR/2015/001047.