On the power graph of a finite group

Abstract

The power graph P(G) of a group G is the graph whose vertex set is the group elements and two elements are adjacent if one is a power of the other. In this paper, we consider some graph theoretical properties of a power graph P(G) that can be related to its group theoretical properties. As consequences of our results, simple proofs for some earlier results are presented.

Vertex connectivity of the power graph of a finite cyclic group

“…In this paper, we consider G = C n , a finite cyclic group of order n and study the vertex connectivity κ(P(C n )) of P(C n ). We note that the graph P(C n ) has the maximum number of edges among all the power graphs of finite groups of order n. This property of P(C n ) was conjectured by Mirzargar et al in [12] and proved by Curtin and Pourgholi in [6]. So it is natural to expect that κ(P(C n )) would be large in general.…”

Edge-maximality of power graphs of finite cyclic groups

“…In this paper, we resolve in the affirmative a conjecture of Mirzargar et al [13,Conjecture 2] concerning the number of edges in the power graph of a finite group. Motivated by the work of Kelarev and Quinn [9,10,11,12], Chakrabarty, Ghosh, and Sen [8] introduced undirected power graphs to study semigroups and groups.…”

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

“…By [13,Theorem 5] and [15,Corollary 2.5], the power graph of a finite group is perfect, in particular, the clique number and the chromatic number coincide. Explicit formula for the clique number of the power graph of a finite cyclic group is given in [20,Theorem 2] and [13,Theorem 7].…”