Maximum spectral gaps of graphs

IF 1.1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2026-02-01 Epub Date: 2025-10-24 DOI:10.1016/j.laa.2025.10.024

George Brooks , William Linz , Linyuan Lu

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引用次数: 0

Abstract

The spread of a graph G is the difference

λ_{1} - λ_{n}

between the largest and smallest eigenvalues of its adjacency matrix. Breen, Riasanovsky, Tait and Urschel recently determined the graph on n vertices with maximum spread for sufficiently large n. In this paper, we study a related question of maximizing the difference

λ_{i + 1} - λ_{n - j}

for a given pair

(i, j)

over all graphs on n vertices. We give upper bounds for all pairs

(i, j)

, exhibit an infinite family of pairs where the bound is tight, and show that for the pair

(1, 0)

the extremal example is unique. These results contribute to a line of inquiry pioneered by Nikiforov aiming to maximize different linear combinations of eigenvalues over all graphs on n vertices.

查看原文本刊更多论文

图的最大谱隙

图G的扩展是其邻接矩阵的最大特征值和最小特征值之间的差λ1−λn。Breen, Riasanovsky， Tait和Urschel最近确定了n个顶点上的图在足够大的n下具有最大的扩展。在本文中，我们研究了给定对（i,j）在所有n个顶点上的差值λi+1−λn−j最大化的相关问题。我们给出了所有对（i,j）的上界，展示了一个无限族的紧界对，并证明了对（1,0）的极值例子是唯一的。这些结果有助于Nikiforov开创的一系列研究，旨在最大化n个顶点上所有图的特征值的不同线性组合。

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.