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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Bibliographic Details
Published in:Discrete mathematics Vol. 348; no. 4; p. 114382
Main Authors: Dalfó, C., Fiol, M.A., Messegué, A.
Format: Journal Article
Language:English
Published: Elsevier B.V 01.04.2025
Subjects:
Algebraic connectivity
Binomial matrix
Laplacian spectrum
Token graph
Algebraic connectivity
Token graph
Binomial matrix
Laplacian spectrum
ISSN:0012-365X
Online Access:Get full text
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Summary: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