Gain distance matrices for complex unit gain graphs

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi=(G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. %A complex unit gain graph($\mathbb{T}$-gain graph) is a simple graph where each orientation of an edge is given a complex unit, and its inverse is assigned to the opposite orientation of the edge. In this article, we propose gain distance matrices for $\mathbb{T}$-gain graphs. These notions generalize the corresponding known concepts of distance matrices and signed distance matrices. Shahul K. Hameed et al. introduced signed distance matrices and developed their properties. Motivated by their work, we establish several spectral properties, including some equivalences between balanced $\mathbb{T}$-gain graphs and gain distance matrices. Furthermore, we introduce the notion of positively weighted $\mathbb{T}$-gain graphs and study some of their properties. Using these properties, Acharya's and Stani\'c's spectral criteria for balance are deduced. Moreover, the notions of order independence and distance compatibility are studied. Besides, we obtain some characterizations for distance compatibility.