The Main Eigenvalues of the Undirected Power Graph of a Group

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 \jj≠0, where \jj is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group Z n and find a condition on n where P(Z n ) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P(Z n ) where n has a unique prime decomposition n=p r p 2. We also formulate a conjecture on the number of the main eigenvalues of P(Z n ) for an arbitrary positive integer n.