{"title":"The average connectivity matrix of a graph","authors":"Linh Nguyen , Suil O","doi":"10.1016/j.disc.2024.114290","DOIUrl":null,"url":null,"abstract":"<div><div>For a graph <em>G</em> and for two distinct vertices <em>u</em> and <em>v</em>, let <span><math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span> be the maximum number of vertex-disjoint paths joining <em>u</em> and <em>v</em> in <em>G</em>. The average connectivity matrix of an <em>n</em>-vertex connected graph <em>G</em>, written <span><math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is an <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix whose <span><math><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></math></span>-entry is <span><math><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>/</mo><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span> and let <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math></span> be the spectral radius of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any <em>n</em>-vertex connected graph <em>G</em>, we have <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mn>4</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mrow><mi>n</mi></mrow></mfrac></math></span>, which implies a result of Kim and O <span><span>[8]</span></span> stating that for any connected graph <em>G</em>, we have <span><math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mn>2</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, where <span><math><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>u</mi><mo>,</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></msub><mfrac><mrow><mi>κ</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></mfrac></math></span> and <span><math><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the maximum size of a matching in <em>G</em>; equality holds only when <em>G</em> is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely <span><math><mi>ρ</mi><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mover><mrow><mi>κ</mi></mrow><mo>‾</mo></mover></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>≤</mo><mfrac><mrow><mo>(</mo><mi>n</mi><mo>−</mo><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo><mo>(</mo><mn>4</mn><msup><mrow><mi>α</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></span>, and equality in the bound holds only when <em>G</em> is a complete balanced bipartite graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 2","pages":"Article 114290"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-18","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/S0012365X24004217","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
For a graph G and for two distinct vertices u and v, let be the maximum number of vertex-disjoint paths joining u and v in G. The average connectivity matrix of an n-vertex connected graph G, written , is an matrix whose -entry is and let be the spectral radius of . In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any n-vertex connected graph G, we have , which implies a result of Kim and O [8] stating that for any connected graph G, we have , where and is the maximum size of a matching in G; equality holds only when G is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely , and equality in the bound holds only when G is a complete balanced bipartite graph.
对于图 G 和两个不同的顶点 u 和 v,设 κ(u,v) 是连接 G 中 u 和 v 的顶点不相交路径的最大数目。n 个顶点连通图 G 的平均连通性矩阵,记为 Aκ‾(G),是一个 n×n 矩阵,其 (u,v) 项为 κ(u,v)/(n2),设 ρ(Aκ‾(G)) 为 Aκ‾(G) 的谱半径。本文将研究矩阵的一些谱性质。特别是,我们证明了对于任意 n 个顶点的连通图 G,ρ(Aκ‾(G))≤4α′(G)n,这意味着 Kim 和 O [8] 的一个结果,即对于任意连通图 G、κ‾(G)≤2α′(G),其中κ‾(G)=∑u,v∈V(G)κ(u,v)(n2),α′(G)是 G 中匹配的最大大小;只有当 G 是具有奇数个顶点的完整图时,相等关系才成立。此外,对于二叉图,我们改进了边界,即 ρ(Aκ‾(G))≤(n-α′(G))(4α′(G)-2)n(n-1) ,只有当 G 是完整的平衡二叉图时,边界相等才成立。
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.