{"title":"Techniques for determining equality of the maximum nullity and the zero forcing number of a graph","authors":"Derek Young","doi":"10.13001/ELA.2021.4967","DOIUrl":null,"url":null,"abstract":"It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters which bound the maximum nullity of a graph below. In particular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdi\\'ere type parameter and the vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"20 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Combinatorics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.13001/ELA.2021.4967","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
It is known that the zero forcing number of a graph is an upper bound for the maximum nullity of the graph. In this paper, we search for characteristics of a graph that guarantee the maximum nullity of the graph and the zero forcing number of the graph are the same by studying a variety of graph parameters which bound the maximum nullity of a graph below. In particular, we introduce a new graph parameter which acts as a lower bound for the maximum nullity of the graph. As a result, we show that the Aztec Diamond graph's maximum nullity and zero forcing number are the same. Other graph parameters that are considered are a Colin de Verdi\'ere type parameter and the vertex connectivity. We also use matrices, such as a divisor matrix of a graph and an equitable partition of the adjacency matrix of a graph, to establish a lower bound for the nullity of the graph's adjacency matrix.
已知图的零强迫数是图的最大零值的上界。本文通过研究限定图的最大零值的各种图参数,寻找保证图的最大零值与图的强制零值相同的特征。特别地,我们引入了一个新的图参数作为图的最大零值的下界。结果表明,阿兹特克菱形图的最大零值和零强迫数是相同的。考虑的其他图形参数包括Colin de Verdi\'ere类型参数和顶点连通性。我们还使用矩阵,如图的除数矩阵和图的邻接矩阵的公平划分,来建立图的邻接矩阵的零的下界。