{"title":"Graph Limits and Spectral Extremal Problems for Graphs","authors":"Lele Liu","doi":"10.1137/22m1508807","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 590-608, March 2024. <br/> 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].","PeriodicalId":49530,"journal":{"name":"SIAM Journal on Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1508807","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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].
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.
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