On -gain graphs with few positive eigenvalues

Abstract

Let ${\mathbb{T}}_4=\{1,-1,\mathbf{i},-\mathbf{i}\}$ be the group of fourth roots of unit. A ${\mathbb{T}}_4$-gain graph is a graph where each orientation of an edge is given a complex unit in ${\mathbb{T}}_4$, which is the inverse of the complex unit assigned to the opposite orientation. In this paper, we characterize the structure of the ${\mathbb{T}}_4$-gain graphs with exactly one positive eigenvalue and determine the ${\mathbb{T}}_4$-gain graphs with cut vertices having exactly two positive eigenvalues. Our results extend some parallel ones about mixed graphs and signed graphs.

KEYWORDS:

• ${\mathbb{T}}_4$-gain graph
• mixed graph
• signed graph
• spectrum

AMS SUBJECT CLASSIFICATION:

• 05C50

Acknowledgments

The authors are so grateful to the referee for their valuable comments and corrections which improve the presentation of the paper.

Disclosure statement

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

Additional information

Funding

This research was supported by National Natural Science Foundation of China (NSFC) [grant numbers 11671402, 11871479, 12001544, 12271527], Natural Science Foundation of Hunan Province [grant numbers 2016JJ2138, 2018JJ2479, 2021JJ40707], the Fundamental Research Funds for the Central Universities of Central South University [grant number 2021zzts0034].