The Main Eigenvalues of the Undirected Power Graph of a Group

Abstract

The undirected power graph of a finite group G, P (G), is a graph with the group elements of G as vertices and two vertices are adjacent if and only if one of them is a power of the other. Let A be an adjacency matrix of P (G). An eigenvalue λ of A is a main eigenvalue if the eigenspace ε(λ) has an eigenvector X such that X t j ̸ = 0, where j is the all-one vector.In this paper we want to focus on the power graph of the finite cyclic group Zn and find a condition on n where P (Zn) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P (Zn) where n has a unique prime decomposition n = p r p2. We also formulate a conjecture on the number of the main eigenvalues of P (Zn) for an arbitrary positive integer n.