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On two algebras of token graphs

M. A. Reyes, C. Dalfó, M. A. Fiol

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 article, we describe some properties of the Laplacian matrix $Ł_k$ of $F_k(G)$ and the Laplacian matrix $\overlineŁ_k$ of the $k$-token graph $F_k(\overline{G})$ of its complement $\overline{G}$. In this context, a result about the commutativity of the matrices $Ł_k$ and $\overlineŁ_k$ was given in [C. Dalfó, F. Duque, R. Fabila-Monroy, M. A. Fiol, C. Huemer, A. L. Trujillo-Negrete, and F. J. Zaragoza Martínez, On the Laplacian spectra of token graphs, {\em Linear Algebra Appl.} {\bf 625} (2021) 322--348], but the proof was incomplete, and there were some typos. Here, we give the correct proof. Based on this result, and fixed the pair $(n,k)$ and the graph $G$, we first introduce a `local' algebra ${\cal L}(G)$, generated by the pair $(Ł_k, \overlineŁ_k)$, showing its closed relationship with the Bose-Mesner algebra of the Johnson graphs $J(n,k)$. Finally, fixed only $(n,k)$, we present a `global' algebra ${\cal A}(n,k)$ that contains ${\cal L}(G)$ together with the Laplacian and adjacency matrices of the $k$-token graph of any graph $G$ on $n$ vertices.

On two algebras of token graphs

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 article, we describe some properties of the Laplacian matrix of and the Laplacian matrix of the -token graph of its complement . In this context, a result about the commutativity of the matrices and was given in [C. Dalfó, F. Duque, R. Fabila-Monroy, M. A. Fiol, C. Huemer, A. L. Trujillo-Negrete, and F. J. Zaragoza Martínez, On the Laplacian spectra of token graphs, {\em Linear Algebra Appl.} {\bf 625} (2021) 322--348], but the proof was incomplete, and there were some typos. Here, we give the correct proof. Based on this result, and fixed the pair and the graph , we first introduce a `local' algebra , generated by the pair , showing its closed relationship with the Bose-Mesner algebra of the Johnson graphs . Finally, fixed only , we present a `global' algebra that contains together with the Laplacian and adjacency matrices of the -token graph of any graph on vertices.
Paper Structure (9 sections, 12 theorems, 24 equations, 4 figures, 3 tables)

This paper contains 9 sections, 12 theorems, 24 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and background
  3. Token graphs
  4. Graph spectra and orthogonal polynomials
  5. Distance-regular graphs and the Bose-Mesner algebra
  6. Johnson graphs
  7. On the Laplacian spectra of token graphs
  8. A 'local' algebra of token graphs
  9. A 'global' algebra of token graphs

Key Result

Proposition 2.1

Let $G$ be a graph with adjacency matrix $\hbox{\boldmath $A$}$ with $d+1$ distinct eigenvalues and Laplacian matrix $\hbox{\boldmath $L$}$ with $l+1$ distinct eigenvalues. Let $p_0,\ldots,p_d$ be its predistance polynomials, and $q_0,\ldots,q_l$ its Laplacian predistance polynomials. Consider the s

Figures (4)

  • Figure 1: The Johnson graph $J(5,2)=F_2(K_5)=\overline{P}$.
  • Figure 2: The graphs $G$, its complement $\overline{G}$, $K_4$, and their 2-token graphs.
  • Figure 3: A graph on 6 vertices and its complement.
  • Figure 4: The Laplacian polynomials of the matrix $\hbox{\boldmath $R$}$.

Theorems & Definitions (23)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 3.1: ddffhtz21
  • Proposition 3.2: ddffhtz21
  • proof
  • Corollary 3.3
  • Proposition 3.4: df22
  • Example 3.5
  • Example 3.6
  • Lemma 3.7
  • ...and 13 more