{"title":"A note proving the nullity of block graphs is unbounded","authors":"","doi":"10.1016/j.disc.2024.114289","DOIUrl":null,"url":null,"abstract":"<div><div>Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24004205","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
Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a -free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of -free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.