{"title":"关于有禁止子图的图的补集的零强制数","authors":"","doi":"10.1016/j.laa.2024.09.009","DOIUrl":null,"url":null,"abstract":"<div><p>Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on <em>n</em> vertices is either <span><math><mi>n</mi><mo>−</mo><mn>3</mn></math></span> or <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.</p></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the zero forcing number of the complement of graphs with forbidden subgraphs\",\"authors\":\"\",\"doi\":\"10.1016/j.laa.2024.09.009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on <em>n</em> vertices is either <span><math><mi>n</mi><mo>−</mo><mn>3</mn></math></span> or <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.</p></div>\",\"PeriodicalId\":18043,\"journal\":{\"name\":\"Linear Algebra and its Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Linear Algebra and its Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0024379524003719\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Linear Algebra and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0024379524003719","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
零强迫和最大无效性是两个重要的图参数，为了帮助解决逆特征值问题，我们对它们进行了大量研究。观察到 n 个顶点上的树的补集的零强制数是 n-3 或 n-1（在一种特殊情况下），这在一定程度上激励了我们，我们考虑了在某些条件下更一般的图的补集的零强制数，特别是那些不包含完整双方子图的图。我们的研究还远远超出了树的范围，完全研究了单环图和仙人掌图补集的所有可能的零强制数。最后，我们得出了所考虑的几组图补集的最大无效数和零强制数之间的相等关系。
On the zero forcing number of the complement of graphs with forbidden subgraphs
Zero forcing and maximum nullity are two important graph parameters which have been laboriously studied in order to aid in the resolution of the Inverse Eigenvalue problem. Motivated in part by an observation that the zero forcing number for the complement of a tree on n vertices is either or in one exceptional case, we consider the zero forcing number for the complement of more general graphs under certain conditions, particularly those that do not contain complete bipartite subgraphs. We also move well beyond trees and completely study all of the possible zero forcing numbers for the complements of unicyclic graphs and cactus graphs. Finally, we yield equality between the maximum nullity and zero forcing number of several families of graph complements considered.
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.
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