{"title":"On the Difference Between the Skew-rank of an Oriented Graph and the Rank of Its Underlying Graph","authors":"Jia-min Zhu, Bo-jun Yuan, Yi Wang","doi":"10.1007/s10255-024-1103-x","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a simple graph and <i>G</i><sup><i>σ</i></sup> be the oriented graph with <i>G</i> as its underlying graph and orientation <i>σ</i>. The rank of the adjacency matrix of <i>G</i> is called the rank of <i>G</i> and is denoted by <i>r</i>(<i>G</i>). The rank of the skew-adjacency matrix of <i>G</i><sup><i>σ</i></sup> is called the skew-rank of <i>G</i><sup><i>σ</i></sup> and is denoted by <i>sr</i>(<i>G</i><sup><i>σ</i></sup>). Let <i>V</i>(<i>G</i>) be the vertex set and <i>E</i>(<i>G</i>) be the edge set of <i>G</i>. The cyclomatic number of <i>G</i>, denoted by <i>c</i>(<i>G</i>), is equal to ∣<i>E</i>(<i>G</i>)∣ − ∣<i>V</i>(<i>G</i>)∣+ <i>ω</i>(<i>G</i>), where <i>ω</i>(<i>G</i>) is the number of the components of <i>G</i>. It is proved for any oriented graph <i>G</i><sup><i>σ</i></sup> that −2<i>c</i>(<i>G</i>) ⩽ sr(<i>G</i><sup><i>σ</i></sup>) − <i>r</i>(<i>G</i>) ⩽ 2<i>c</i>(<i>G</i>). In this paper, we prove that there is no oriented graph <i>G</i><sup><i>σ</i></sup> with <i>sr</i>(<i>G</i><sup><i>σ</i></sup>) − <i>r</i>(<i>G</i>) = 2<i>c</i>(<i>G</i>)−1, and in addition, there are in nitely many oriented graphs <i>G</i><sup><i>σ</i></sup> with connected underlying graphs such that <i>c</i>(<i>G</i>) = <i>k</i> and <i>sr</i>(<i>G</i><sup><i>σ</i></sup>)−<i>r</i>(<i>G</i>) = 2<i>c</i>(<i>G</i>)−ℓ for every integers <i>k</i>, ℓ satisfying 0 ⩽ ℓ ⩽ 4<i>k</i> and ℓ≠ 1.</p></div>","PeriodicalId":6951,"journal":{"name":"Acta Mathematicae Applicatae Sinica, English Series","volume":"40 1","pages":"129 - 136"},"PeriodicalIF":0.9000,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematicae Applicatae Sinica, English Series","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10255-024-1103-x","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Let G be a simple graph and Gσ be the oriented graph with G as its underlying graph and orientation σ. The rank of the adjacency matrix of G is called the rank of G and is denoted by r(G). The rank of the skew-adjacency matrix of Gσ is called the skew-rank of Gσ and is denoted by sr(Gσ). Let V(G) be the vertex set and E(G) be the edge set of G. The cyclomatic number of G, denoted by c(G), is equal to ∣E(G)∣ − ∣V(G)∣+ ω(G), where ω(G) is the number of the components of G. It is proved for any oriented graph Gσ that −2c(G) ⩽ sr(Gσ) − r(G) ⩽ 2c(G). In this paper, we prove that there is no oriented graph Gσ with sr(Gσ) − r(G) = 2c(G)−1, and in addition, there are in nitely many oriented graphs Gσ with connected underlying graphs such that c(G) = k and sr(Gσ)−r(G) = 2c(G)−ℓ for every integers k, ℓ satisfying 0 ⩽ ℓ ⩽ 4k and ℓ≠ 1.
期刊介绍:
Acta Mathematicae Applicatae Sinica (English Series) is a quarterly journal established by the Chinese Mathematical Society. The journal publishes high quality research papers from all branches of applied mathematics, and particularly welcomes those from partial differential equations, computational mathematics, applied probability, mathematical finance, statistics, dynamical systems, optimization and management science.