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 xtj ≠ 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 = pr p2. we also formulate a conjecture on the number of the main eigenvalues of p (zn) for an arbitrary positive integer n.