On the spectra of token graphs of cycles and other graphs
The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) e...
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| Vydáno v: | Linear algebra and its applications Ročník 679; s. 38 - 66 |
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15.12.2023
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| Abstract | The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn). |
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| AbstractList | The k-token graph Fk(G) of a graph G is the graph whose vertices are the k-subsets of vertices from G, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is a known result that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity of G. In this paper, we first give results that relate the algebraic connectivities of a token graph and the same graph after removing a vertex. Then, we prove the result on the algebraic connectivity of 2-token graphs for two infinite families: the odd graphs Or for all r, and the multipartite complete graphs Kn1,n2,…,nr for all n1,n2,…,nr In the case of cycles, we present a new method that allows us to compute the whole spectrum of F2(Cn). This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of F2(Cn). |
| Author | Dalfó, C. Messegué, A. Reyes, M.A. Fiol, M.A. |
| Author_xml | – sequence: 1 givenname: M.A. surname: Reyes fullname: Reyes, M.A. email: monicaandrea.reyes@udl.cat organization: Dept. de Matemàtica, Universitat de Lleida, Lleida/Igualada, Catalonia – sequence: 2 givenname: C. orcidid: 0000-0002-8438-9353 surname: Dalfó fullname: Dalfó, C. email: cristina.dalfo@udl.cat organization: Dept. de Matemàtica, Universitat de Lleida, Lleida/Igualada, Catalonia – sequence: 3 givenname: M.A. surname: Fiol fullname: Fiol, M.A. email: miguel.angel.fiol@upc.edu organization: Dept. de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Catalonia – sequence: 4 givenname: A. surname: Messegué fullname: Messegué, A. email: visitant.arnau.messegue@udl.cat organization: Dept. de Matemàtica, Universitat de Lleida, Lleida/Igualada, Catalonia |
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| Cites_doi | 10.1063/1.5084136 10.21136/CMJ.1973.101168 10.1080/03081087.2022.2042174 10.1016/j.disc.2009.10.011 10.1080/00927872.2014.975349 10.1007/s003730200055 10.5614/ejgta.2017.5.2.10 10.1007/BF01396012 10.1016/j.laa.2021.05.005 10.1007/s10801-018-0862-y 10.1080/00150517.1999.12428871 10.1137/0611016 10.1016/j.jctb.2006.04.002 10.1016/j.dam.2018.10.040 10.1007/s00373-011-1055-9 |
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| Keywords | 05C10 05C50 Laplacian spectrum Algebraic connectivity Regular partition Token graph Binomial matrix Lift graph 05C15 |
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| SubjectTerms | Algebraic connectivity Binomial matrix Laplacian spectrum Lift graph Regular partition Token graph |
| Title | On the spectra of token graphs of cycles and other graphs |
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