{"title":"On the incidence matrix of a graph with matrix weights","authors":"Madhab Mondal, Sukanta Pati, Bhaba Kumar Sarma","doi":"10.1016/j.laa.2025.09.003","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>G</em> be a simple, oriented, edge weighted graph with <em>n</em> vertices and <em>m</em> edges, where weights are matrices in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> (the set of all square matrices of order <em>s</em>). It is well-known that if <em>G</em> is connected and weights are nonzero scalars, then the rank of the <em>vertex-edge incidence matrix</em> <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. We observe that if the weights are rank <em>k</em> matrices in <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span>, then <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>k</mi><mo>≤</mo><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>min</mi><mo></mo><mo>{</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mi>s</mi><mo>,</mo><mi>m</mi><mi>k</mi><mo>}</mo></math></span>. In particular, when <span><math><mi>k</mi><mo>=</mo><mn>1</mn></math></span> (i.e., weights as rank one matrices) and <span><math><mi>s</mi><mo>≥</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span>, then <span><math><mi>n</mi><mo>−</mo><mn>1</mn><mo>≤</mo><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>m</mi></math></span>. We show that for large values of <em>s</em>, there exist assignments of rank one weights from <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>s</mi></mrow></msub></math></span> such that all integer values between <span><math><mi>n</mi><mo>−</mo><mn>1</mn></math></span> and <em>m</em> can be attained as the rank of <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. Further, we study the smallest possible values of <em>s</em> for which these ranks can be attained. Surprisingly, the smallest value of <em>s</em> for which <em>m</em> can be achieved as <span><math><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> need not be <span><math><mo>⌈</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></mfrac><mo>⌉</mo></math></span>. Even more interestingly, it turns out that the minimum value of <em>s</em> is the arboricity of the graph, i.e., the least number of colors needed to color the edges of <em>G</em> so that no cycle is monochromatic. As an extension, we supply an expression of the minimum values of <em>s</em> for which the intermediate values for the ranks of <span><math><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can be achieved. For a graph <em>G</em>, <span><math><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can give more information on <em>G</em> if we consider matrix weights. We show that a connected graph <em>G</em> on <em>n</em> vertices is a tree if and only if for every assignment of rank one weights from <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mi>rank</mi><mspace></mspace><mi>Q</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></math></span>. Such characterization is not available when we consider nonzero scalar weights. We also establish a characterization for unicyclic graphs based on the rank of the incidence matrix. Moreover, we discuss briefly the Laplacian matrices of graphs with matrix weights in view of the results obtained in this paper.</div></div>","PeriodicalId":18043,"journal":{"name":"Linear Algebra and its Applications","volume":"728 ","pages":"Pages 295-319"},"PeriodicalIF":1.1000,"publicationDate":"2025-09-08","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/S0024379525003696","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
Abstract
Let G be a simple, oriented, edge weighted graph with n vertices and m edges, where weights are matrices in (the set of all square matrices of order s). It is well-known that if G is connected and weights are nonzero scalars, then the rank of the vertex-edge incidence matrix is . We observe that if the weights are rank k matrices in , then . In particular, when (i.e., weights as rank one matrices) and , then . We show that for large values of s, there exist assignments of rank one weights from such that all integer values between and m can be attained as the rank of . Further, we study the smallest possible values of s for which these ranks can be attained. Surprisingly, the smallest value of s for which m can be achieved as need not be . Even more interestingly, it turns out that the minimum value of s is the arboricity of the graph, i.e., the least number of colors needed to color the edges of G so that no cycle is monochromatic. As an extension, we supply an expression of the minimum values of s for which the intermediate values for the ranks of can be achieved. For a graph G, can give more information on G if we consider matrix weights. We show that a connected graph G on n vertices is a tree if and only if for every assignment of rank one weights from , . Such characterization is not available when we consider nonzero scalar weights. We also establish a characterization for unicyclic graphs based on the rank of the incidence matrix. Moreover, we discuss briefly the Laplacian matrices of graphs with matrix weights in view of the results obtained in this paper.
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.