{"title":"The Jacobian of a graph and graph automorphisms","authors":"","doi":"10.1016/j.disc.2024.114259","DOIUrl":null,"url":null,"abstract":"<div><p>In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph <em>X</em> in the group of symmetries of the Jacobian of <em>X</em>. As a consequence we show that if a 3-edge-connected graph <em>X</em> admits a nonabelian semiregular group of automorphisms, then the Jacobian of <em>X</em> cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of <em>X</em> is well-understood – it is equal to the number of spanning trees of <em>X</em> – the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-11","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/S0012365X2400390X","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
In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph X in the group of symmetries of the Jacobian of X. As a consequence we show that if a 3-edge-connected graph X admits a nonabelian semiregular group of automorphisms, then the Jacobian of X cannot be cyclic. In particular, Cayley graphs of degree at least three arising from nonabelian groups have non-cyclic Jacobians. While the size of the Jacobian of X is well-understood – it is equal to the number of spanning trees of X – the combinatorial interpretation of the rank of Jacobian of a graph is unknown. Our paper presents a contribution in this direction.
在本文中，我们研究了图 X 的自变量群的某些线性表示在 X 的雅各布对称群中的忠实性。结果表明，如果一个三边连接的图 X 接受一个非阿贝尔半圆自变量群，那么 X 的雅各布不可能是循环的。特别是，由非阿贝尔群产生的阶数至少为 3 的 Cayley 图具有非循环雅各布。虽然 X 的雅各布的大小很好理解--它等于 X 的生成树的数量，但对图的雅各布秩的组合解释还不清楚。我们的论文在这方面做出了贡献。
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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