A general method to find the spectrum and eigenspaces of the $k$-token of a cycle, and 2-token through continuous fractions
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913290476584960 |
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| author | Reyes, M. A. Dalfó, C. Fiol, M. A. Messegué, A. |
| author_facet | Reyes, M. A. Dalfó, C. Fiol, M. A. Messegué, 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 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 $C_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_20148 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A general method to find the spectrum and eigenspaces of the $k$-token of a cycle, and 2-token through continuous fractions Reyes, M. A. Dalfó, C. Fiol, M. A. Messegué, 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 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 $C_n$. |
| title | A general method to find the spectrum and eigenspaces of the $k$-token of a cycle, and 2-token through continuous fractions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2403.20148 |