{"title":"On the algebraic connectivity of some token graphs","authors":"C. Dalfó, M. A. Fiol","doi":"10.1007/s10801-024-01323-0","DOIUrl":null,"url":null,"abstract":"<p>The <i>k</i>-token graph <span>\\(F_k(G)\\)</span> of a graph <i>G</i> is the graph whose vertices are the <i>k</i>-subsets of vertices from <i>G</i>, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in <i>G</i>. It was proved that the algebraic connectivity of <span>\\(F_k(G)\\)</span> equals the algebraic connectivity of <i>G</i> with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of <span>\\(F_k(G)\\)</span> equals the one of <i>G</i> for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.</p>","PeriodicalId":14926,"journal":{"name":"Journal of Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebraic Combinatorics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10801-024-01323-0","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The k-token graph \(F_k(G)\) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It was proved that the algebraic connectivity of \(F_k(G)\) equals the algebraic connectivity of G with a proof using random walks and interchange of processes on a weighted graph. However, no algebraic or combinatorial proof is known, and it would be a hit in the area. In this paper, we algebraically prove that the algebraic connectivity of \(F_k(G)\) equals the one of G for new infinite families of graphs, such as trees, some graphs with hanging trees, and graphs with minimum degree large enough. Some examples of these families are the following: the cocktail party graph, the complement graph of a cycle, and the complete multipartite graph.
图 G 的 k 标记图 \(F_k(G)\)是指其顶点是来自 G 的顶点的 k 子集的图，只要它们的对称差是 G 中的一对相邻顶点，其中的两个顶点就是相邻的。有人证明了 \(F_k(G)\)的代数连通性等于 G 的代数连通性，证明中使用了加权图上的随机行走和交换过程。然而，目前还没有代数或组合证明，这将是该领域的一个重大突破。在本文中，我们用代数方法证明了对于新的无限图族，如树、一些有悬挂树的图和最小度足够大的图，\(F_k(G)\)的代数连通性等于 G 的代数连通性。这些族的一些例子如下：鸡尾酒会图、循环补图和完整多方图。
期刊介绍：
The Journal of Algebraic Combinatorics provides a single forum for papers on algebraic combinatorics which, at present, are distributed throughout a number of journals. Within the last decade or so, algebraic combinatorics has evolved into a mature, established and identifiable area of mathematics. Research contributions in the field are increasingly seen to have substantial links with other areas of mathematics.
The journal publishes papers in which combinatorics and algebra interact in a significant and interesting fashion. This interaction might occur through the study of combinatorial structures using algebraic methods, or the application of combinatorial methods to algebraic problems.
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