On the spectrum of complex unit gain graph

A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called $\mathbb{T}$-gain adjacency matrix. Let $\mathbb{T}_{G}$ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. In this article, we study the cospectrality of matrices in $\mathbb{T}_{G}$ and we establish equivalent conditions for a graph $G$ to be a tree in terms of the spectrum and the spectral radius of matrices in $\mathbb{T}_{G}$. We identify a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G \in \mathfrak{F^{'}}$, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of $\mathbb{T}$-gain adjacency matrices on $G \in \mathfrak{F^{'}}$ in terms of their largest eigenvalues. Thereupon, we characterize $\mathbb{T}$-gain graphs for which the spectral radius of the associated $\mathbb{T}$-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result 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.].