Spectral properties of distance Laplacian matrices of complex unit gain graphs

Abstract

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. In this article, we study several spectral properties of distance Laplacian matrices of $\mathbb{T}$-gain graphs. In particular, we establish a characterization for the balanced $ \mathbb{T}$-gain graph in terms of the nullity of gain distance Laplacian matrices. As an example, it is shown that two switching equivalent $ \mathbb{T} $-gain graphs need not imply that their distance Laplacian spectra are the same. However, we provide a necessary condition for which two switching equivalent $ \mathbb{T} $-gain graphs have the same distance Laplacian spectra. Furthermore, we present a lower bound for spectral radii of gain distance Laplacian matrices in terms of the winner index. In addition, we establish some upper bounds for spectral radii of gain distance Laplacian matrices and characterize the equalities.