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IMCS/Publications/CSJM/Issues/CSJM v.29, n.1 (85), 2021/

Total energy of signed digraphs

Authors: Sumaira Hafeez, Mehtab Khan
Keywords: Signed Digraphs, Total energy, Increasing prop- erty, T-equienergetic sidigraphs.

Abstract

The energy of a sidigraph is defined as the sum of absolute values of real parts of its eigenvalues. The iota energy of a sidigraph is defined as the sum of absolute values of imaginary parts of its eigenvalues. Recently a new notion of energy of digraphs is introduced which is called the total energy of digraphs. In this paper, we extend this concept of total energy to sidigraphs. We compute total energy formulas for negative directed cycles and show that the total energy of negative directed cycles with fixed order increases monotonically. We introduce complex adjacency matrix to give the integral representation for total energy of sidigraphs. We discuss the increasing property of total energy over some particular subfamilies of $S_{n,h}$, where $S_{n,h}$ contains $n$-vertex sidigraphs with each cycle having length $h$. Using the Cauchy-Schwarz inequality, we find upper bound for the total energy of sidigraphs. Finally, we find the class of noncospectral equienergetic sidigraphs.

Sumaira Hafeez
School of Natural Sciences,
National University of Sciences and Technology,
H-12 Islamabad, Pakistan
E-mail:

Mehtab Khan
School Mathematical Sciences, Anhui University,
Hefei, China
E-mail:

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