Daniel A. Jaume , Diego G. Martinez , Cristian Panelo
{"title":"Determinantal decomposition of graphs with a unique perfect matching","authors":"Daniel A. Jaume , Diego G. Martinez , Cristian Panelo","doi":"10.1016/j.disc.2024.114370","DOIUrl":null,"url":null,"abstract":"<div><div>In this work, a structural decomposition of graphs with a unique perfect matching is introduced. The decomposition is given by the barbell subgraphs: even subdivisions of two <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> graphs joined by an edge such that the unique perfect matching of <em>G</em> induces a perfect matching in the subgraphs. The decomposition breaks a graph <em>G</em>, with a unique perfect matching, into two subgraphs, one of which is a Kőnig-Egerváry graph. Furthermore, the decomposition is shown to be multiplicative with respect to determinantal-type (Schur) functions of the adjacency matrix of graphs with a unique perfect matching. Additionally, in this work, Godsil's formula for the determinant of trees with perfect matching is extended to all graphs with a unique perfect matching.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 4","pages":"Article 114370"},"PeriodicalIF":0.7000,"publicationDate":"2025-04-01","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/S0012365X24005016","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2024/12/27 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
In this work, a structural decomposition of graphs with a unique perfect matching is introduced. The decomposition is given by the barbell subgraphs: even subdivisions of two graphs joined by an edge such that the unique perfect matching of G induces a perfect matching in the subgraphs. The decomposition breaks a graph G, with a unique perfect matching, into two subgraphs, one of which is a Kőnig-Egerváry graph. Furthermore, the decomposition is shown to be multiplicative with respect to determinantal-type (Schur) functions of the adjacency matrix of graphs with a unique perfect matching. Additionally, in this work, Godsil's formula for the determinant of trees with perfect matching is extended to all graphs with a unique perfect matching.
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.
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