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 \lambda of A is a main eigenvalue if the eigenspace \epsilon(\lambda) has an eigenvector X such that X^{t}\jj\neq ۰, where \jj is the all-one vector. In this paper we want to focus on the power graph of the finite cyclic group \mathbb{Z}_{n} and find a condition on n where P(\mathbb{Z}_{n}) has exactly one main eigenvalue. Then we calculate the number of main eigenvalues of P(\mathbb{Z}_{n}) where n has a unique prime decomposition n = p^{r} p_۲. We also formulate a conjecture on the number of the main eigenvalues of P(\mathbb{Z}_{n}) for an arbitrary positive integer n.

Keywords:

Power graph , Main eigenvalue , Cyclic group , Prime divisor