about the sharpness of the bound in terms of largest vertex degree established in \[Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs. J. Graph Theory 85 (2017), no. 1, 217-248.\].'
author:
- 'Aniruddha Samanta[^1]'
- 'M. Rajesh Kannan[^2]'
bibliography:
- 'raj-ani-ref1.bib'
title: On the spectrum of complex unit gain graph
---

[**AMS Subject Classification(2010):**]{} 05C50, 05C22, 15B57.

**Keywords.** Gain graph, Suitably oriented graph, Normal spanning tree, Distinguished edge property, Perron-Frobenius theory, Spectral radius.

Introduction
============

For a given group $\mathfrak{G}$, a $\mathfrak{G}$-gain graph is a graph $G$ with each orientation of an edge of $G$ is assigned an element $g \in \mathfrak{G}$ (called gain of the oriented edge) and whose inverse $g^{-1}$ is assigned to the opposite orientation of the edge. For more details about the notion of $\mathfrak{G}$-gain graphs, we refer to [@Zaslav-gain-balance; @zas4; @Zaslav; @Zaslav-biased-graph-II; @Zaslav-biased-graph-III; @Zaslav-biased-graph-IV]. Let $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\} $ be the multiplicative group of unit complex numbers. In [@reff1], the notion of $\mathbb{T}$-gain graphs has been studied. If $G$ is a graph with vertex set $ V(G)=\{v_{1}, v_{2}, \dots, v_{n} \} $ and each orientation of its edges having some gain from $\mathbb{T}$, then the associated $\mathbb{T}$-gain adjacency matrix is an $n \times 