复杂单位增益图的无效性界限

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-07-14 DOI:10.1016/j.laa.2024.07.006

Qian-Qian Chen , Ji-Ming Guo

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引用次数: 0

摘要

复数单位增益图或 T 增益图是一个三元组 Φ=(G,T,φ)，由作为 Φ 底图的简单图 G、单位复数集合 T={z∈C:|z|=1} 和增益函数 φ:E→→T 组成，其性质为 φ(eij)=φ(eji)-1 。本文首先证明不存在空性为 n(G)-2m(G)+2c(G)-1（其中 n(G)、m(G)和 c(G) 分别为 G 的阶、匹配数和循环数）的复数单位增益图。最后，我们描述了所有非星状复数单位增益双方形仙人掌图的特征，这概括了 Wong 等人（2022）[30] 的一个结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Bounds of nullity for complex unit gain graphs

A complex unit gain graph, or $T$ -gain graph, is a triple $Φ = (G, T, φ)$ comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers $T = {z \in C : | z | = 1}$ , and a gain function $φ : \vec{E} \to T$ with the property that $φ (e_{i j}) = φ {(e_{j i})}^{- 1}$ . A cactus graph is a connected graph in which any two cycles have at most one vertex in common.

In this paper, we firstly show that there does not exist a complex unit gain graph with nullity $n (G) - 2 m (G) + 2 c (G) - 1$ , where $n (G)$ , $m (G)$ and $c (G)$ are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30].

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.