On the Laplacian spectra of token graphs

TL;DR

This paper investigates the Laplacian spectra of token graphs, revealing spectral relationships between different token graphs of a given graph and identifying specific families with coinciding algebraic connectivities.

Contribution

It establishes spectral inclusion relations between token graphs of different sizes and connects spectra of a graph and its complement, extending known eigenvalue properties.

Findings

Spectral inclusion: Laplacian spectrum of F_h(G) is contained in that of F_k(G) for h ≤ k.

Double odd and doubled Johnson graphs are token graphs of complete and star graphs.

Algebraic connectivities of certain token graphs and original graphs coincide.

Abstract

We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. 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 $G$. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers $h$ and $k$ such that $1\le h\le k\le \frac{n}{2}$, the Laplacian spectrum of $F_h(G)$ is contained in the Laplacian spectrum of $F_k(G)$. We also show that the double odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph $K_n$ and the star $S_{n}=K_{1,n-1}$, respectively. Besides, we obtain a relationship between the spectra of the $k$-token graph of $G$ and the $k$-token graph of its complement $\overline{G}$. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs. Finally, the double odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph $G$ and its token graph.