Ivan I. Kyrchei , Eran Treister , Volodymyr O. Pelykh
{"title":"四元数单位增益图邻接矩阵的行列式","authors":"Ivan I. Kyrchei , Eran Treister , Volodymyr O. Pelykh","doi":"10.1016/j.disc.2025.114659","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper, we present a combinatorial description of the determinant of the adjacency matrix of a quaternion unit gain graph using recently introduced row-column noncommutative determinants by one of the authors. We define a quaternion unit gain graph as a graph in which each edge's orientation is assigned a quaternion unit, and its opposite orientation is assigned the inverse of this quaternion unit. Initially, we provide detailed combinatorial descriptions of the determinants of the adjacency matrices for a single cycle and a path graph with quaternion unit gains. Subsequently, we investigate the determinant of the adjacency matrix for quaternion unit gain graphs whose underlying graphs consist of multiple cycles and/or path graphs. We introduce a decomposition procedure for such graphs involving reductions obtained by cutting off edges associated with branch vertices so that each reduction's adjacency matrix is equal to the direct sum of its components' adjacency matrices. Our resulting theorem offers a combinatorial description for obtaining the determinant of an adjacency matrix in terms of cycle and graph path adjacency determinants on which they are decomposed. The obtained results are novel for quaternion unit gain graphs and complex ones, and they could be applied to various types of gain graphs, not just those with unit gains.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 1","pages":"Article 114659"},"PeriodicalIF":0.7000,"publicationDate":"2025-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The determinant of the adjacency matrix of a quaternion unit gain graph\",\"authors\":\"Ivan I. Kyrchei , Eran Treister , Volodymyr O. Pelykh\",\"doi\":\"10.1016/j.disc.2025.114659\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper, we present a combinatorial description of the determinant of the adjacency matrix of a quaternion unit gain graph using recently introduced row-column noncommutative determinants by one of the authors. We define a quaternion unit gain graph as a graph in which each edge's orientation is assigned a quaternion unit, and its opposite orientation is assigned the inverse of this quaternion unit. Initially, we provide detailed combinatorial descriptions of the determinants of the adjacency matrices for a single cycle and a path graph with quaternion unit gains. Subsequently, we investigate the determinant of the adjacency matrix for quaternion unit gain graphs whose underlying graphs consist of multiple cycles and/or path graphs. We introduce a decomposition procedure for such graphs involving reductions obtained by cutting off edges associated with branch vertices so that each reduction's adjacency matrix is equal to the direct sum of its components' adjacency matrices. Our resulting theorem offers a combinatorial description for obtaining the determinant of an adjacency matrix in terms of cycle and graph path adjacency determinants on which they are decomposed. The obtained results are novel for quaternion unit gain graphs and complex ones, and they could be applied to various types of gain graphs, not just those with unit gains.</div></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"349 1\",\"pages\":\"Article 114659\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2025-07-04\",\"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/S0012365X25002675\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X25002675","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
The determinant of the adjacency matrix of a quaternion unit gain graph
In this paper, we present a combinatorial description of the determinant of the adjacency matrix of a quaternion unit gain graph using recently introduced row-column noncommutative determinants by one of the authors. We define a quaternion unit gain graph as a graph in which each edge's orientation is assigned a quaternion unit, and its opposite orientation is assigned the inverse of this quaternion unit. Initially, we provide detailed combinatorial descriptions of the determinants of the adjacency matrices for a single cycle and a path graph with quaternion unit gains. Subsequently, we investigate the determinant of the adjacency matrix for quaternion unit gain graphs whose underlying graphs consist of multiple cycles and/or path graphs. We introduce a decomposition procedure for such graphs involving reductions obtained by cutting off edges associated with branch vertices so that each reduction's adjacency matrix is equal to the direct sum of its components' adjacency matrices. Our resulting theorem offers a combinatorial description for obtaining the determinant of an adjacency matrix in terms of cycle and graph path adjacency determinants on which they are decomposed. The obtained results are novel for quaternion unit gain graphs and complex ones, and they could be applied to various types of gain graphs, not just those with unit gains.
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.