Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

M. A. Reyes, C. Dalfó, M. A. Fiol, A. Messegué

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 $C_n$.

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

Abstract

The -token graph of a graph is the graph whose vertices are the -subsets of vertices from , two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in . In this paper, we propose a general method to find the spectrum and eigenspaces of the -token graph of a cycle . The method is based on the theory of lift graphs and the recently introduced theory of over-lifts. In the case of , we use continuous fractions to derive the spectrum and eigenspaces of the 2-token graph of .
Paper Structure (6 sections, 7 theorems, 30 equations, 6 figures, 5 tables)

This paper contains 6 sections, 7 theorems, 30 equations, 6 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Lift graphs and over-lifts
  3. A general method for computing the spectrum of $F_k(C_n)$
  4. The spectrum of $F_2(C_n)$ through continuous fractions
  5. Some examples
  6. The characteristic polynomials of $F_2(C_n)$

Key Result

Theorem 2.1

Let $R(n)$ be the set of $n$-th roots of unity, and consider the base graph $G=(V,E)$ with voltage assignment $\alpha$ with the cyclic group ${\cal G}=\mathbb Z_n$. If $\hbox{\boldmath $x$}=(x_u)_{u\in V}$ is an eigenvector of $\hbox{\boldmath $B$}(z)$ with eigenvalue $\lambda$, then the vector $\hb

Figures (6)

  • Figure 1: The 2-token and 3-token graphs of the cycle $C_7$.
  • Figure 2: The base graph $G$ for the 3-token graph of $C_7$.
  • Figure 3: The 3-token graph of $C_6$.
  • Figure 4: The characteristic polynomials of $\hbox{\boldmath $B$}(r)$, with $r=0,1,2$, for $F_2(C_5)$.
  • Figure 5: The characteristic polynomial of $\hbox{\boldmath $B$}(1)$, with smallest root being the algebraic connectivity of $F_2(C_9)$.
  • ...and 1 more figures

Theorems & Definitions (15)

  • Theorem 2.1: dfmrs17
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Proposition 4.1: rdfm23
  • Theorem 4.2
  • ...and 5 more