The eigenvalue multiplicity of line graphs

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-09-03 DOI:10.1016/j.laa.2024.08.021

Sarula Chang , Jianxi Li , Yirong Zheng

引用次数: 0

Abstract

Let $m (G, λ)$ , $c (G)$ and $p (G)$ be the multiplicity of an eigenvalue λ, the cyclomatic number and the number of pendant vertices of a connected graph G, respectively. Yang et al. (2023) [10] proved that $m (L (T), λ) \leq p (T) - 1$ for any tree T, and characterized all trees T with $m (L (T), λ) = p (T) - 1$ , where $L (T)$ is the line graph of T. In this paper, we extend their result from a tree T to any graph $G \neq C_{n}$ , and prove that $m (L (G), λ) \leq 2 c (G) + p (G) - 1$ for any graph $G \neq C_{n}$ . Moreover, all graphs G with $m (L (G), - 1) = 2 c (G) + p (G) - 1$ are completely characterized.

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线图的特征值多重性

设 m(G,λ)、c(G) 和 p(G) 分别为连通图 G 的特征值 λ 的倍率、循环数和挂顶点数。Yang 等人（2023）[10] 证明了对于任意树 T，m(L(T),λ)≤p(T)-1，并表征了 m(L(T),λ)=p(T)-1 的所有树 T，其中 L(T) 是 T 的线图。本文将他们的结果从树 T 扩展到任何图 G≠Cn，并证明对于任何图 G≠Cn，m(L(G),λ)≤2c(G)+p(G)-1。此外，m(L(G),-1)=2c(G)+p(G)-1 的所有图形 G 都是完全表征的。

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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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.