The spectrum of symmetric decorated paths

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-02-10 DOI:10.1016/j.laa.2025.02.011

Gabriel Coutinho, Emanuel Juliano, Thomás Jung Spier

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引用次数: 0

Abstract

The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported on the end vertices of the path so that their difference is less than the square root of two in the even distance case, and less than one in the odd distance case. As a first application, we show that these end vertices cannot be involved in a quantum walk phenomenon known as perfect state transfer, significantly strengthening a recent result by two of the authors along with Godsil and van Bommel. For a second application, we show that there is no balanced integral tree of odd diameter greater than three, answering a question raised by Híc and Nedela in 1998.

Our main technique involves manipulating ratios of characteristic polynomials of graphs and subgraphs into continued fractions, and exploring in detail their analytic properties. We will also make use of a result due to Pólya and Szegö about functions that preserve the Lebesgue measure, which as far as we know is a novel application to combinatorics. In the end, we connect our machinery to a recently introduced algorithm to locate eigenvalues of trees, and with our approach we show that any graph which contains two vertices separated by a unique path that is the subdivision of a bridge with at least six inner vertices cannot be integral. As a minor corollary this implies that most trees are not integral, but we believe no one thought otherwise.

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对称装饰路径的谱

本文的主要结果表明，在以某种镜像对称方式配置的有根图的路径的根积中，在路径的端点上支持不同的特征值，使得它们的差值在偶数距离情况下小于根号2，在奇数距离情况下小于1。作为第一个应用，我们证明了这些端点不能参与量子行走现象，即完美状态转移，这大大加强了Godsil和van Bommel最近的两个作者的结果。对于第二个应用，我们证明了不存在奇径大于3的平衡积分树，回答了Híc和Nedela在1998年提出的问题。我们的主要技术包括将图和子图的特征多项式的比率处理为连分式，并详细探索它们的解析性质。我们还将利用Pólya和Szegö关于保留勒贝格测度的函数的结果，据我们所知，这是组合学中的一个新应用。最后，我们将我们的机器连接到最近引入的一种算法来定位树的特征值，并且通过我们的方法，我们证明了任何包含两个顶点的图都不能是积分的，这两个顶点是由至少有六个内顶点的桥的细分的唯一路径分开的。作为一个次要的推论，这意味着大多数树不是整体的，但我们相信没有人会这样认为。

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.