The power graph of ℤ_n with at most two main signless Laplacian eigenvalues

Abstract

Let G be a finite group. The power graph of G is a graph with vertices G and two vertices are adjacent if one is the power of another. Suppose Q be a signless laplacian matrix of a power graph of G. The main signlesslaplacian eigenvalue of G is an eigenvalue of Q that has an eigenvector x which the sum of its entries is non-zero. In this paper we consider the power graph of a finite cyclic group ℤ_n, P(ℤ_n), and find n for if P(ℤ_n) has exactly one and two main signlesslaplacian eigenvalues.