Gain distance Laplacian matrices for complex unit gain graphs

2024/04/25 by Suliman Khan, Khan, Suliman · 1 citation
Computer Science · Mathematics · #05C05 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2404.17085

openalex publication_date 2024/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A complex unit gain graph (or a \mathbbT-gain graph) Θ(Σ,φ) is a graph where the unit complex number is assign by a function φ to every oriented edge of Σ and assign its inverse to the opposite orientation. In this paper, we define the two gain distance Laplacian matrices DL^max_<(Θ) and DL^min_<(Θ) corresponding to the two gain distance matrices D^max_<(Θ) and D^min_<(Θ) defined for \mathbbT-gain graphs Θ(Σ,φ), for any vertex ordering (V(Σ),<). 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 \mathbbT-gain graphs.

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• Spectral properties of distance Laplacian matrices of complex unit gain graphs

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