On the spectra and spectral radii of token graphs
Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G), has as vertices the (n/k) k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn) is the Johnson graph J(...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2024 |
| País: | España |
| Recursos: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositório: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:10459.1/464851 |
| Acesso em linha: | https://doi.org/10.1007/s40590-023-00583-3 https://hdl.handle.net/10459.1/464851 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Token graph Adjacency spectrum Local spectrum Laplacian spectrum Algebraic connectivity Binomial matrix Spectral radius Walk-regular graph |
| Resumo: | Let G be a graph on n vertices. The k-token graph (or symmetric k-th power) of G, denoted by Fk(G), has as vertices the (n/k) k-subsets of vertices from G, and two vertices are adjacent when their symmetric difference is a pair of adjacent vertices in G. In particular, Fk(Kn) is the Johnson graph J(n, k), which is a distance-regular graph used in coding theory. In this paper, we present some results concerning the (adjacency and Laplacian) spectrum of Fk(G) in terms of the spectrum of G. For instance, when G is walk-regular, an exact value for the spectral radius (or maximum eigenvalue) of Fk(G) is obtained. When G is distance-regular, other eigenvalues of its 2-token graph are derived using the theory of equitable partitions. A generalization of Aldous’ spectral gap conjecture (which is now a theorem) is proposed. |
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