{"title":"Algebraic Connectedness and Bipartiteness of Quantum Graphs","authors":"Junichiro Matsuda","doi":"10.1007/s00220-024-05046-y","DOIUrl":null,"url":null,"abstract":"<p>Connectedness and bipartiteness are basic properties of classical graphs, and the purpose of this paper is to investigate the case of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms. This paper shows that regular tracial quantum graphs have the same algebraic characterization of connectedness and bipartiteness as classical graphs. We also prove the equivalence between bipartiteness and two-colorability of quantum graphs by comparing two notions of graph homomorphisms: one respects adjacency matrices and the other respects edge spaces. In particular, all kinds of quantum two-colorability are mutually equivalent for regular connected tracial quantum graphs.</p>","PeriodicalId":522,"journal":{"name":"Communications in Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s00220-024-05046-y","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
Connectedness and bipartiteness are basic properties of classical graphs, and the purpose of this paper is to investigate the case of quantum graphs. We introduce the notion of connectedness and bipartiteness of quantum graphs in terms of graph homomorphisms. This paper shows that regular tracial quantum graphs have the same algebraic characterization of connectedness and bipartiteness as classical graphs. We also prove the equivalence between bipartiteness and two-colorability of quantum graphs by comparing two notions of graph homomorphisms: one respects adjacency matrices and the other respects edge spaces. In particular, all kinds of quantum two-colorability are mutually equivalent for regular connected tracial quantum graphs.
期刊介绍:
The mission of Communications in Mathematical Physics is to offer a high forum for works which are motivated by the vision and the challenges of modern physics and which at the same time meet the highest mathematical standards.