On the Automorphisms of Token Graphs Generated by 2-cuts with the Same Neighbours

Let G be a connected graph on n vertices and 1 ≤ k ≤ n-1 1 ≤ k ≤ n - 1 an integer. The k -token graph of G is the graph Fₖ(G) F k ( G ) , whose vertices are all the k -subsets of vertices of G , two of which are adjacent whenever their symmetric difference is an edge of G . Every automorphism of G induces an automorphism of Fₖ(G) F k ( G ) in a natural way. Suppose that S:=,y\ S : = { x , y } is a cut set of G , such that x and y have the same neighbours in G ,y\ G \ { x , y } . In this paper, we show that there exists a large number of automorphisms of Fₖ(G) F k ( G ) defined by S that are not induced by automorphisms of G . We also describe the group produced by all such 2-cuts of G .