特征值倍数为 n - d 的最小图形

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2024-06-17 DOI:10.1016/j.laa.2024.06.015

Yuanshuai Zhang , Dein Wong , Wenhao Zhen

引用次数: 0

摘要

对于阶数为 n 的连通图 G，让 e(G) 表示其不同特征值的个数，d 表示直径。众所周知，e(G)≥d+1，这表明对于任意实数 μ，mG(μ)≤n-d。如果 e(G)=d+1 ，则称为最小图。2013 年，Wong 等人描述了 mG(0)=n-d 的所有最小图。在本文中，我们运用星形补码理论证明，如果 G 不是路径，且 mG(μ)=n-d 时，则 μ∈{0,-1}。此外，我们完全描述了所有 mG(-1)=n-d 的最小图。

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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Minimal graphs with eigenvalue multiplicity of n − d

For a connected graph G with order n, let $e (G)$ be the number of its distinct eigenvalues and d be the diameter. We denote by $m_{G} (μ)$ the eigenvalue multiplicity of μ in G. It is well known that $e (G) \geq d + 1$ , which shows $m_{G} (μ) \leq n - d$ for any real number μ. A graph is called $m i n i m a l$ if $e (G) = d + 1$ . In 2013, Wong et al. characterize all minimal graphs with $m_{G} (0) = n - d$ . In this paper, by applying the star complement theory, we prove that if G is not a path and $m_{G} (μ) = n - d$ , then $μ \in {0, - 1}$ . Furthermore, we completely characterize all minimal graphs with $m_{G} (- 1) = n - d$ .

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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.