Spectral properties of token graphs
Let G be a graph on n vertices. For a given integer k such that 1≤k≤n, the k-token graph Fk(G) of G is defined as the graph whose vertices are the k-subsets of the vertex set of G, and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article,...
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| Vydáno v: | Linear algebra and its applications Ročník 687; s. 181 - 206 |
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Elsevier Inc
15.04.2024
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| ISSN: | 0024-3795, 1873-1856 |
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| Abstract | Let G be a graph on n vertices. For a given integer k such that 1≤k≤n, the k-token graph Fk(G) of G is defined as the graph whose vertices are the k-subsets of the vertex set of G, and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of Fk(G) and obtain bounds on the adjacency and Laplacian spectral radii of Fk(G). Interestingly, it is found that Sn, the star graph on n vertices, has the same Laplacian spectral radius as that of Fk(Sn). It was conjectured that for any graph G, the algebraic connectivity of Fk(G) is equal to the algebraic connectivity of G. This result turned out to be a theorem, as it was proved by using the theory of the continuous Markov chain of random walks and the interchange process. However, proving this theorem using algebraic and combinatorial methods is still an open and interesting problem. Using combinatorial techniques, we prove that the theorem holds for a class of graphs that have a cut-vertex of degree n−1. We also prove it by restricting the smallest degree of the k-token graph of G. |
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| AbstractList | Let G be a graph on n vertices. For a given integer k such that 1≤k≤n, the k-token graph Fk(G) of G is defined as the graph whose vertices are the k-subsets of the vertex set of G, and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of Fk(G) and obtain bounds on the adjacency and Laplacian spectral radii of Fk(G). Interestingly, it is found that Sn, the star graph on n vertices, has the same Laplacian spectral radius as that of Fk(Sn). It was conjectured that for any graph G, the algebraic connectivity of Fk(G) is equal to the algebraic connectivity of G. This result turned out to be a theorem, as it was proved by using the theory of the continuous Markov chain of random walks and the interchange process. However, proving this theorem using algebraic and combinatorial methods is still an open and interesting problem. Using combinatorial techniques, we prove that the theorem holds for a class of graphs that have a cut-vertex of degree n−1. We also prove it by restricting the smallest degree of the k-token graph of G. |
| Author | Verma, Piyush Barik, Sasmita |
| Author_xml | – sequence: 1 givenname: Sasmita orcidid: 0000-0002-3927-5218 surname: Barik fullname: Barik, Sasmita email: sasmita@iitbbs.ac.in – sequence: 2 givenname: Piyush surname: Verma fullname: Verma, Piyush email: s21ma09010@iitbbs.ac.in |
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| Cites_doi | 10.1103/PhysRevLett.85.5468 10.1007/s003730200055 10.1016/j.laa.2004.01.020 10.1080/03081080600679029 10.1090/S0894-0347-10-00659-4 10.1016/j.dam.2018.03.085 10.1038/35065725 10.1016/0024-3795(95)00199-2 10.1063/1.5084136 10.7151/dmgt.1959 10.1016/j.laa.2021.05.005 10.1016/j.jctb.2010.07.001 10.1016/j.physa.2014.02.043 10.1137/S0895480191222653 10.26493/2590-9770.1244.720 10.21136/CMJ.1973.101168 10.1016/j.jctb.2006.04.002 10.1016/j.laa.2006.08.017 10.1007/s00373-011-1055-9 |
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| Keywords | 05C10 05C75 05C50 Laplacian spectrum Spectral radius Algebraic connectivity Token graph Adjacency spectrum |
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| SubjectTerms | Adjacency spectrum Algebraic connectivity Laplacian spectrum Spectral radius Token graph |
| Title | Spectral properties of token graphs |
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