Gain distance Laplacian matrices for complex unit gain graphs

TL;DR

This paper introduces new gain distance Laplacian matrices for complex unit gain graphs, characterizes their singularity and rank, and explores balanced graphs, extending results to weighted gain graphs.

Contribution

It defines two new gain distance Laplacian matrices for $ ext{T}$-gain graphs and provides their singularity, rank formulas, and characterizations of balanced graphs, generalizing to weighted cases.

Findings

Defined two gain distance Laplacian matrices for $ ext{T}$-gain graphs.

Provided formulas for the rank and conditions for singularity of these matrices.

Characterized balanced complex unit gain graphs using the new Laplacian matrices.

Abstract

A complex unit gain graph (or a $\mathbb{T}$-gain graph) $\Theta(\Sigma,\varphi)$ is a graph where the unit complex number is assign by a function $\varphi$ to every oriented edge of $\Sigma$ and assign its inverse to the opposite orientation. In this paper, we define the two gain distance Laplacian matrices $DL^{\max}_{<}(\Theta)$ and $DL^{\min}_{<}(\Theta)$ corresponding to the two gain distance matrices $D^{\max}_{<}(\Theta)$ and $D^{\min}_{<}(\Theta)$ defined for $\mathbb{T}$-gain graphs $\Theta(\Sigma,\varphi)$, for any vertex ordering $(V(\Sigma),<)$. Furthermore, we provide the characterization of singularity and find formulas for the rank of those Laplacian matrices. We also establish two types of characterization for balanced in complex unit gain graphs while using the gain distance Lapalcian matrices. Most of the results are derived by proving them more generally for weighted $\mathbb{T}$-gain graphs.