On structural and spectral properties of reduced power graph of finite groups

Abstract

The reduced power graph of a finite group [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the elements of the group [Formula: see text] and two vertices [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. In this paper, we first describe the structure of the reduced power graph of the finite cyclic group [Formula: see text]. Consequently, we provide a short and alternative proof of one of the results published in (R. Rajkumar and T. Anitha, Laplacian spectrum of reduced power graph of certain finite groups, Linear Multilinear Algebra (2019) 1–18). We characterize the values of [Formula: see text] for which [Formula: see text] is a line graph. We then deduce the signless Laplacian spectrum of [Formula: see text] using its structure. We provide lower and upper bounds on the signless Laplacian spectral radius of [Formula: see text]. Finally, we conclude the paper by determining the signless Laplacian spectrum of [Formula: see text], where [Formula: see text] denotes the dihedral group of order [Formula: see text].