Geodesic
Well-covered token graphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 767-792

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The k-token graph T_k(G) is the graph whose vertices are the k-subsets of vertices of a graph G, with two vertices of T_k(G) adjacent if their symmetric difference is an edge of G. We explore when T_k(G) is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs G, we classify when T_k(G) is well-covered. For an arbitrary graph G, we show that if T_2(G) is well-covered, then the girth of G is at most four. We include upper and lower bounds on the independence number of T_k(G), and provide some families of well-covered token graphs.
Keywords: independence number, well-covered graph, token graph, double vertex graph, symmetric power of a graph 
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     title = {Well-covered token graphs},
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Abdelmalek, F.M.; Vander Meulen, Esther; Vander Meulen, Kevin N.; Van Tuyl, Adam. Well-covered token graphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 3, pp. 767-792. http://geodesic.mathdoc.fr/item/DMGT_2023_43_3_a12/