{"title":"关于具有给定阶数和广义 4-independence 数的图的谱极值","authors":"","doi":"10.1016/j.amc.2024.129018","DOIUrl":null,"url":null,"abstract":"<div><p>Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph <em>G</em>, a vertex subset <em>S</em> is called a maximum generalized 4-independent set of <em>G</em> if the induced subgraph <span><math><mi>G</mi><mo>[</mo><mi>S</mi><mo>]</mo></math></span> dose not contain a 4-tree as its subgraph, and the subset <em>S</em> has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of <em>G</em>. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the <em>n</em>-vertex graphs having the minimum spectral radius with generalized 4-independence number <em>ψ</em>, where <span><math><mi>ψ</mi><mo>⩾</mo><mrow><mo>⌈</mo><mn>3</mn><mi>n</mi><mo>/</mo><mn>4</mn><mo>⌉</mo></mrow></math></span>. Finally, we identify all the connected <em>n</em>-vertex graphs with generalized 4-independence number <span><math><mi>ψ</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mrow><mo>⌈</mo><mn>3</mn><mi>n</mi><mo>/</mo><mn>4</mn><mo>⌉</mo></mrow><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>}</mo></math></span> having the minimum spectral radius.</p></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":null,"pages":null},"PeriodicalIF":3.5000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S009630032400479X/pdfft?md5=d1bd990f782f3cab80b3cb783b379e1a&pid=1-s2.0-S009630032400479X-main.pdf","citationCount":"0","resultStr":"{\"title\":\"On spectral extrema of graphs with given order and generalized 4-independence number\",\"authors\":\"\",\"doi\":\"10.1016/j.amc.2024.129018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph <em>G</em>, a vertex subset <em>S</em> is called a maximum generalized 4-independent set of <em>G</em> if the induced subgraph <span><math><mi>G</mi><mo>[</mo><mi>S</mi><mo>]</mo></math></span> dose not contain a 4-tree as its subgraph, and the subset <em>S</em> has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of <em>G</em>. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the <em>n</em>-vertex graphs having the minimum spectral radius with generalized 4-independence number <em>ψ</em>, where <span><math><mi>ψ</mi><mo>⩾</mo><mrow><mo>⌈</mo><mn>3</mn><mi>n</mi><mo>/</mo><mn>4</mn><mo>⌉</mo></mrow></math></span>. Finally, we identify all the connected <em>n</em>-vertex graphs with generalized 4-independence number <span><math><mi>ψ</mi><mo>∈</mo><mo>{</mo><mn>3</mn><mo>,</mo><mrow><mo>⌈</mo><mn>3</mn><mi>n</mi><mo>/</mo><mn>4</mn><mo>⌉</mo></mrow><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>}</mo></math></span> having the minimum spectral radius.</p></div>\",\"PeriodicalId\":55496,\"journal\":{\"name\":\"Applied Mathematics and Computation\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.5000,\"publicationDate\":\"2024-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S009630032400479X/pdfft?md5=d1bd990f782f3cab80b3cb783b379e1a&pid=1-s2.0-S009630032400479X-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics and Computation\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S009630032400479X\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics and Computation","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S009630032400479X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
给定图类中具有最大或最小谱半径的图的特征是谱极值图论中的一个经典问题,最初由 Brualdi 和 Solheid 提出。在给定图 G 的情况下,如果诱导子图 G[S] 的子图中不包含 4 树,并且子集 S 具有最大的心数,那么顶点子集 S 就被称为 G 的最大广义 4-independent 集。在本文中,我们首先确定了在所有具有固定阶数和广义 4-independence 数的连通图(也称双方图、树)中具有最大谱半径的连通图(也称双方图、树),此外,我们还建立了具有固定阶数的树的广义 4-independence 数的下界。其次,我们描述了具有最小谱半径、广义 4-independence 数为 ψ(其中 ψ⩾⌈3n/4⌉)的所有 n 顶点图的结构。最后,我们确定了具有最小谱半径的广义 4-independence 数ψ∈{3,⌈3n/4⌉,n-1,n-2}的所有 n 顶点连通图。
On spectral extrema of graphs with given order and generalized 4-independence number
Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where . Finally, we identify all the connected n-vertex graphs with generalized 4-independence number having the minimum spectral radius.
期刊介绍:
Applied Mathematics and Computation addresses work at the interface between applied mathematics, numerical computation, and applications of systems – oriented ideas to the physical, biological, social, and behavioral sciences, and emphasizes papers of a computational nature focusing on new algorithms, their analysis and numerical results.
In addition to presenting research papers, Applied Mathematics and Computation publishes review articles and single–topics issues.