Vertex connectivity of the power graph of a finite cyclic group II

摘要

The power graph P(G) of a given finite group G is the simple undirected graph whose vertices are the elements of G, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity kappa(P(G)) of P(G) is the minimum number of vertices which need to be removed from O so that the induced subgraph of P(G) on the remaining vertices is disconnected or has only one-vertex. For a positive integer n, let C-n be the cyclic group of order n. Suppose that the prime power decomposition of n is given by n = p(1)(n1)p(2)(n2) center dot center dot center dot p(r)(nr). where r >= 1, n(1), n(2), n(r), are positive integers and p(1), p(2), ..., p(r) are prime numbers with p(1) < p(2) < ... < p(r). The vertex connectivity kappa(P(C-n)) of P(C-n) is known for r <= 3, see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.20,18.06.0011. In this paper, for r >= 4, we give a new upper bound for kappa(P(C-n)) and determine kappa(P(C-n)) when n(r) >= 2. We also determine kappa(P(C-n) when n is a product of distinct prime numbers.