Some bounds on the Laplacian eigenvalues of token graphs

The k-token graph Fk(G) of a graph G on n vertices is the graph whose vertices are the (nk)k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is known that the algebraic connectivity (or second Laplacian eigenvalue) of...

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Vydané v:Discrete mathematics Ročník 348; číslo 4; s. 114382
Hlavní autori: Dalfó, C., Fiol, M.A., Messegué, A.
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier B.V 01.04.2025
Predmet:
Algebraic connectivity
Binomial matrix
Laplacian spectrum
Token graph
Algebraic connectivity
Token graph
Binomial matrix
Laplacian spectrum
ISSN:0012-365X
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Popis
Shrnutí:The k-token graph Fk(G) of a graph G on n vertices is the graph whose vertices are the (nk)k-subsets of vertices from G, two of which are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. It is known that the algebraic connectivity (or second Laplacian eigenvalue) of Fk(G) equals the algebraic connectivity α(G) of G. In this paper, we give some bounds on the (Laplacian) eigenvalues of the k-token graph (including the algebraic connectivity) in terms of the h-token graph, with h≤k. For instance, we prove that if λ is an eigenvalue of Fk(G), but not of G, thenλ≥kα(G)−k+1. As a consequence, we conclude that if α(G)≥k, then α(Fh(G))=α(G) for every h≤k.
ISSN:0012-365X
DOI:10.1016/j.disc.2024.114382