{"title":"Algebraic degrees of quasi-abelian semi-Cayley digraphs","authors":"","doi":"10.1016/j.disc.2024.114178","DOIUrl":null,"url":null,"abstract":"<div><p>For a digraph Γ, if <em>F</em> is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of Γ, then <em>F</em> is called the splitting field of Γ. The extension degree of <em>F</em> over the field of rational numbers <span><math><mi>Q</mi></math></span> is said to be the algebraic degree of Γ. A digraph is a semi-Cayley digraph over a group <em>G</em> if it admits <em>G</em> as a semiregular automorphism group with two orbits of equal size. A semi-Cayley digraph <span><math><mrow><mi>SC</mi></mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>12</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>21</mn></mrow></msub><mo>)</mo></math></span> is called quasi-abelian if each of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>22</mn></mrow></msub><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>12</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mn>21</mn></mrow></msub></math></span> is a union of some conjugacy classes of <em>G</em>. This paper determines the splitting field and the algebraic degree of a quasi-abelian semi-Cayley digraph over any finite group in terms of irreducible characters of groups. This work generalizes the previous works on algebraic degrees of Cayley graphs over abelian groups and any group having a subgroup of index 2, and semi-Cayley digraphs over abelian groups.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-22","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/S0012365X24003091","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 digraph Γ, if F is the smallest field that contains all roots of the characteristic polynomial of the adjacency matrix of Γ, then F is called the splitting field of Γ. The extension degree of F over the field of rational numbers is said to be the algebraic degree of Γ. A digraph is a semi-Cayley digraph over a group G if it admits G as a semiregular automorphism group with two orbits of equal size. A semi-Cayley digraph is called quasi-abelian if each of and is a union of some conjugacy classes of G. This paper determines the splitting field and the algebraic degree of a quasi-abelian semi-Cayley digraph over any finite group in terms of irreducible characters of groups. This work generalizes the previous works on algebraic degrees of Cayley graphs over abelian groups and any group having a subgroup of index 2, and semi-Cayley digraphs over abelian groups.
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.