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

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 κ (𝒫 (G)) of 𝒫 (G) is the minimum number of vertices which need to be removed from G 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 n 1 p 2 n 2 ⋯ p r n r , 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 κ (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.2018.06.001]. In this paper, for r ≥ 4 , we give a new upper bound for κ (𝒫 (C n)) and determine κ (𝒫 (C n)) when n r ≥ 2. We also determine κ (𝒫 (C n)) when n is a product of distinct prime numbers. [ABSTRACT FROM AUTHOR]