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

Abstract

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], 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 [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], 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.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.