A kind of conditional connectivity of Cayley graphs generated by unicyclic graphs

For a graph G = (V, E), a subset F ⊆ V(G) is called an Rk-vertex-cut of G if G − F is disconnected and each vertex u ∈ V(G) − F has at least k neighbors in G − F. The Rk-vertex-connectivity of G, denoted by κk(G), is the cardinality of the minimum Rk-vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ2 for Cayley graphs generated by unicyclic graphs. Let Sym(n) be the symmetric group on {1, 2, … , n  } and TT be a set of transpositions of Sym(n  ). Let G(T)G(T) be the graph on vertex set {1, 2, … , n  } and edge set {ij:(ij)∈T}{ij:(ij)∈T}. If G(T)G(T) is a unicyclic graph, then simply denote the Cayley graph Cay(Sym(n),T)Cay(Sym(n),T) by UGn. In particular, if G(T)G(T) is a cycle, then UGn is the well-known modified bubble sort graph MBn. We prove in this paper that if m   is the length of the unique cycle in G(T)G(T), thenκ2(UGn)=4n-10,if3=m