给定特征值的树高上限

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2024-02-02 DOI:10.1007/s00493-023-00071-2

Artūras Dubickas

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引用次数: 0

摘要

在本文中，我们证明了每一个度为（d）的全实代数整数（lambda）都会作为某个高度为（d（d+1）/2+3）的树的特征值出现。为了证明这一点，对于一个给定的代数数（α），我们研究一个包含零并且在映射（x）下封闭的可加半群。寻找最小的这样的半群似乎是一个独立的问题。

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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An Upper Bound for the Height of a Tree with a Given Eigenvalue

In this paper we prove that every totally real algebraic integer \(\lambda \) of degree \(d \ge 2\) occurs as an eigenvalue of some tree of height at most \(d(d+1)/2+3\). In order to prove this, for a given algebraic number \(\alpha \ne 0\), we investigate an additive semigroup that contains zero and is closed under the map \(x \mapsto \alpha /(1-x)\) for \(x \ne 1\). The problem of finding the smallest such semigroup seems to be of independent interest.

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来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.