加兰标记图法

IF 1 3区 数学 Q1 MATHEMATICS Linear Algebra and its Applications Pub Date : 2024-07-24 DOI:10.1016/j.laa.2024.07.018
Alan Lew
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Let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <em>G</em>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph <em>G</em> on <em>n</em> vertices and any <span><math><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contained in that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here, we continue to study the relation between the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that, for <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, any eigenvalue <em>λ</em> of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> that is not contained in the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies<span><span><span><math><mi>k</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>≤</mo><mi>k</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is the second smallest eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also known as the algebraic connectivity of <em>G</em>), and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is its largest eigenvalue. 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Let <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <em>G</em>, and <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> be the Laplacian matrix of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph <em>G</em> on <em>n</em> vertices and any <span><math><mn>0</mn><mo>≤</mo><mi>ℓ</mi><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is contained in that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>.</p><p>Here, we continue to study the relation between the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and that of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In particular, we show that, for <span><math><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mrow><mo>⌊</mo><mi>n</mi><mo>/</mo><mn>2</mn><mo>⌋</mo></mrow></math></span>, any eigenvalue <em>λ</em> of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> that is not contained in the spectrum of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> satisfies<span><span><span><math><mi>k</mi><mo>(</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≤</mo><mi>λ</mi><mo>≤</mo><mi>k</mi><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is the second smallest eigenvalue of <span><math><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> (also known as the algebraic connectivity of <em>G</em>), and <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>L</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> is its largest eigenvalue. 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引用次数: 0

摘要

一个图的第-个标记图是这样的图,它的顶点是 的-子集,它的边是所有的-子集对,使得 和 的对称差在 。Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete 和 Zaragoza Martínez 证明,对于任意顶点上的图和任意 , 的谱包含在 的谱中。
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Garland's method for token graphs

The k-th token graph of a graph G=(V,E) is the graph Fk(G) whose vertices are the k-subsets of V and whose edges are all pairs of k-subsets A,B such that the symmetric difference of A and B forms an edge in G. Let L(G) be the Laplacian matrix of G, and Lk(G) be the Laplacian matrix of Fk(G). It was shown by Dalfó, Duque, Fabila-Monroy, Fiol, Huemer, Trujillo-Negrete, and Zaragoza Martínez that for any graph G on n vertices and any 0kn/2, the spectrum of L(G) is contained in that of Lk(G).

Here, we continue to study the relation between the spectrum of Lk(G) and that of Lk1(G). In particular, we show that, for 1kn/2, any eigenvalue λ of Lk(G) that is not contained in the spectrum of Lk1(G) satisfiesk(λ2(L(G))k+1)λkλn(L(G)), where λ2(L(G)) is the second smallest eigenvalue of L(G) (also known as the algebraic connectivity of G), and λn(L(G)) is its largest eigenvalue. Our proof relies on an adaptation of Garland's method, originally developed for the study of high-dimensional Laplacians of simplicial complexes.

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来源期刊
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.
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