On the algebraic connectivity of token graphs and graphs under perturbations

Given a graph G=(V,E) on n vertices and an integer k between 1 and n−1, the k-token graph Fk(G) has vertices representing the k-subsets of V, and two vertices are adjacent if their symmetric difference is the two end-vertices of an edge in E. Using the theory of Markov chains of random walks and the...

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Bibliographic Details
Published in:Discrete Applied Mathematics Vol. 377; pp. 134 - 146
Main Authors: Song, Xiaodi, Dalfó, Cristina, Fiol, Miquel Àngel, Zhang, Shenggui
Format: Journal Article
Language:English
Published: Elsevier B.V 31.12.2025
Subjects:
Algebraic connectivity
Binomial matrix
Cut clique
Kite graph
Laplacian spectrum
Token graph
Algebraic connectivity
Kite graph
Token graph
Binomial matrix
Cut clique
Laplacian spectrum
ISSN:0166-218X
Online Access:Get full text
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Summary:Given a graph G=(V,E) on n vertices and an integer k between 1 and n−1, the k-token graph Fk(G) has vertices representing the k-subsets of V, and two vertices are adjacent if their symmetric difference is the two end-vertices of an edge in E. Using the theory of Markov chains of random walks and the interchange process, it was proved that the algebraic connectivities (second smallest Laplacian eigenvalues) of G and Fk(G) coincide, but a combinatorial/algebraic proof has been shown elusive. In this paper, we use the latter approach and prove that such equality holds for different new classes of graphs under perturbations, such as extended cycles, extended complete bipartite graphs, kite graphs, and graphs with a cut clique. Kite graphs are formed by a graph (head) with several paths (tail) rooted at the same vertex and with exciting properties. For instance, we show that the different eigenvalues of a kite graph are also eigenvalues of its perturbed graph obtained by adding edges. Moreover, as a particular case of one of our theorems, we generalize a recent result of Barik and Verma (2024) about graphs with a cut vertex of degree n−1. Along the way, we give conditions under which the perturbed graph G+uv, with uv∈E, has the same algebraic connectivity as G.
ISSN:0166-218X
DOI:10.1016/j.dam.2025.06.057