A general method to find the spectrum and eigenspaces of the $k$-token of a cycle, and 2-token through continuous fractions

TL;DR

This paper introduces a general method using lift graph theory and continuous fractions to determine the spectrum and eigenspaces of k-token graphs of cycles, providing explicit results for the case k=2.

Contribution

It develops a unified approach to analyze the spectral properties of k-token cycle graphs, including a novel application of continuous fractions for 2-token cases.

Findings

Derived the spectrum and eigenspaces of F_k(C_n) using lift graph theory.

Provided explicit spectral results for the 2-token cycle graph case.

Introduced a new method involving continuous fractions for spectral analysis.

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 $G$. In this paper, we propose a general method to find the spectrum and eigenspaces of the $k$-token graph $F_k(C_n)$ of a cycle $C_n$. The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of $k=2$, we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of $C_n$.