The degree of a vertex in the power graph of a finite abelian group

Abstract

The power graph of a given finite group is a simple undirected graph whose vertex set is the group itself, and there is an edge between any two distinct vertices if one is a power of the other. In this paper, we find a precise formula to count the degree of a vertex in the power graph of a finite abelian group of prime-power order. By using the degree formula, we give a new proof to show that the power graph of a cyclic group of prime-power order is complete. We finally determine the degree of a vertex in the power graph of a finite abelian group.