Graph Limits and Spectral Extremal Problems for Graphs

IF 0.9 3区数学 Q2 MATHEMATICS

SIAM Journal on Discrete Mathematics Pub Date : 2024-01-31 DOI:10.1137/22m1508807

Lele Liu

引用次数: 0

Abstract

SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024.
Abstract. We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let [math] be the largest eigenvalue of the adjacency matrix of a graph [math] and [math] be the complement of [math]. A nice conjecture states that the graph on [math] vertices maximizing [math] is the join of a clique and an independent set with [math] and [math] (also [math] and [math] if [math]) vertices, respectively. We resolve this conjecture for sufficiently large [math] using analytic methods. Our second result concerns the [math]-spread of a graph [math], which is defined as the difference between the largest eigenvalue and least eigenvalue of the signless Laplacian of [math]. It was conjectured by Cvetković, Rowlinson, and Simić [Publ. Inst. Math., 81 (2007), pp. 11–27] that the unique [math]-vertex connected graph of maximum [math]-spread is the graph formed by adding a pendant edge to [math]. We confirm this conjecture for sufficiently large [math].

查看原文本刊更多论文

图形极限和图形谱极值问题

SIAM 离散数学杂志》，第 38 卷，第 1 期，第 590-608 页，2024 年 3 月。摘要。我们证明了谱极值图论中涉及图特征值线性组合的两个猜想。设[math]是图[math]邻接矩阵的最大特征值，[math]是[math]的补集。一个不错的猜想指出，[math] 顶点上[math] 最大的图是一个小群和一个独立集的连接，小群和独立集的顶点分别是[math]和[math]（如果是[math]，则也是[math]和[math]）。对于足够大的 [math]，我们用分析方法解决了这个猜想。我们的第二个结果涉及图[math]的[math]-spread，它被定义为[math]的无符号拉普拉奇的最大特征值和最小特征值之差。根据 Cvetković、Rowlinson 和 Simić 的猜想[Publ. Inst. Math., 81 (2007), pp.对于足够大的 [math]，我们证实了这一猜想。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

SIAM Journal on Discrete Mathematics 数学-应用数学

CiteScore

1.90

自引率

0.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Discrete Mathematics (SIDMA) publishes research papers of exceptional quality in pure and applied discrete mathematics, broadly interpreted. The journal''s focus is primarily theoretical rather than empirical, but the editors welcome papers that evolve from or have potential application to real-world problems. Submissions must be clearly written and make a significant contribution. Topics include but are not limited to: properties of and extremal problems for discrete structures combinatorial optimization, including approximation algorithms algebraic and enumerative combinatorics coding and information theory additive, analytic combinatorics and number theory combinatorial matrix theory and spectral graph theory design and analysis of algorithms for discrete structures discrete problems in computational complexity discrete and computational geometry discrete methods in computational biology, and bioinformatics probabilistic methods and randomized algorithms.