On the spectra of token graphs of cycles and other graphs

TL;DR

This paper investigates the spectral properties of token graphs derived from cycles and other graphs, providing new methods to compute spectra and relating algebraic connectivities before and after vertex removal.

Contribution

It introduces a novel approach to compute the entire spectrum of 2-token graphs of cycles and extends algebraic connectivity results to specific graph families.

Findings

Spectra of 2-token graphs of cycles are explicitly computed.

Asymptotic formulas for eigenvalues of 2-token cycle graphs are derived.

Algebraic connectivity relations are established for token graphs after vertex removal.

Abstract

The $k$-token graph $F_k(G)$ 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 $F_k(G)$ 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 $O_r$ for all $r$, and the multipartite complete graphs $K_{n_1,n_2,\ldots,n_r}$ for all $n_1,n_2,\ldots,n_r$ In the case of cycles, we present a new method that allows us to compute the whole spectrum of $F_2(C_n)$. This method also allows us to obtain closed formulas that give asymptotically exact approximations for most of the eigenvalues of $F_2(\textit{}C_n)$.