Spectral properties of token graphs - Barik, Sasmita; Verma, Piyush (Linear algebra and its applications, 2024)
Let G be a graph on n vertices. For a given integer k such that 1≤k≤n, the k-token graph Fk(G) of G is defined as the graph whose vertices are the k-subsets of the vertex set of G, and two of them are adjacent whenever their symmetric difference is a pair of adjacent vertices in G. In this article, we study the structural and spectral properties of token graphs. We describe the adjacency matrix and the Laplacian matrix of Fk(G) and obtain bounds on the adjacency and Laplacian spectral radii of Fk(G). Interestingly, it is found that Sn, the star graph on n vertices, has the same Laplacian spectral radius as that of Fk(Sn). It was conjectured that for any graph G, the algebraic connectivity of Fk(G) is equal to the algebraic connectivity of G. This result turned out to be a theorem, as it was proved by using the theory of the continuous Markov chain of random walks and the interchange process. However, proving this theorem using algebraic and combinatorial methods is still an open and interesting problem. Using combinatorial techniques, we prove that the theorem holds for a class of graphs that have a cut-vertex of degree n−1. We also prove it by restricting the smallest degree of the k-token graph of G.