Vertex connectivity of the power graph of a finite cyclic group

Abstract

Let n=p1n1p2n2⋯prnr, where r,n1,n2,…,nr are positive integers and p1,p2,…,pr are distinct prime numbers with p1<p2<…<pr. For the finite cyclic group Cn of order n, let P(Cn) be the power graph of Cn and κ(P(Cn)) be the vertex connectivity of P(Cn). It is known that κ(P(Cn))=p1n1−1 if r=1. For r≥2, we determine the exact value of κ(P(Cn)) when 2ϕ(p1p2⋯pr−1)≥p1p2⋯pr−1, and give an upper bound for κ(P(Cn)) when 2ϕ(p1p2⋯pr−1)<p1p2⋯pr−1, which is sharp for many values of n but equality need not hold always.