On two algebras of token graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914730770169856 |
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| author | Reyes, M. A. Dalfó, C. Fiol, M. A. |
| author_facet | Reyes, M. A. Dalfó, C. Fiol, M. A. |
| contents | 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 Mart\'ınez,
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18800 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On two algebras of token graphs Reyes, M. A. Dalfó, C. Fiol, M. A. Combinatorics 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 Mart\'ınez, 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. |
| title | On two algebras of token graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.18800 |