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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Bibliographic Details
Published in:Linear algebra and its applications Vol. 679; pp. 38 - 66
Main Authors: Reyes, M.A., Dalfó, C., Fiol, M.A., Messegué, A.
Format: Journal Article
Language:English
Published: Elsevier Inc 15.12.2023
Subjects:
Algebraic connectivity
Binomial matrix
Laplacian spectrum
Lift graph
Regular partition
Token graph
05C10
05C50
Laplacian spectrum
Algebraic connectivity
Regular partition
Token graph
Binomial matrix
Lift graph
05C15
ISSN:0024-3795, 1873-1856
Online Access:Get full text
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Summary: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).
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2023.09.004