Bounds for the rank of a complex unit gain graph in terms of the independence number

2019/09/17 by Shengjie He, Rong‐Xia Hao, He, Shengjie +3
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph Labeling and Dimension Problems #Graph theory and applications

paper · pdf · doi:10.48550/arxiv.1909.08533

openalex publication_date 2019/09/17 · openalex created_date 2019/09/26 · openalex updated_date 2026/07/28

Abstract

A complex unit gain graph (or \mathbbT-gain graph) is a triple Φ=(G, \mathbbT, φ) ((G, φ) for short) consisting of a graph G as the underlying graph of (G, φ), \mathbbT= \ z ∈ C:|z|=1 \ is a subgroup of the multiplicative group of all nonzero complex numbers ℂ^× and a gain function φ: \overrightarrowE → \mathbbT such that φ(e_ij)=φ(e_ji)^-1=φ(e_ji). In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph (G, φ) with order n, and prove that 2n-2c(G) ≤ r(G, φ)+2α(G) ≤ 2n. Where r(G, φ), α(G) and c(G) are the rank of the Hermitian adjacency matrix A(G, φ), the independence number and the cyclomatic number of G, respectively. Furthermore, the properties of the complex unit gain graph that reaching the lower bound are characterized.

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