Vertex connectivity of the power graph of a finite cyclic group

Let n = p(1)(n1)p(2)(n2) . . . p(r)(nr), where r, n(1), n(2), . . . , n(r) are positive integers and p(1), p(2), . . . , P-r are distinct prime numbers with p(1) < p(2) < . . . < p(r). For the finite cyclic group C-n, of order n, let P(C-n) be the power graph of C-n and kappa(P(C-n)) be the vertex connectivity of P(C-n). It is known that kappa(P(C-n)) = p(1)(n1) - 1 if r = 1. For r >= 2, we determine the exact value of kappa(P(C-n)) when 2 phi(p(1)p(2) . . . Pr-1) >= P1P2 . . . Pr-1, and give an upper bound for kappa(P(C-n)) when 2 phi(p(1)p(2) . . . Pr-1) < p(1)p(2) . . . Pr-1, which is sharp for many values of n but equality need not hold always. (C) 2018 Elsevier B.V. All rights reserved.