Laplacian spectra of Coprime Graph of finite cyclic and Dihedral groups

Abstract

Let [Formula: see text] denote the Coprime Graph of a finite group [Formula: see text]. In this paper we study the Laplacian eigenvalues of the Coprime Graph of the finite cyclic group [Formula: see text] and the Dihedral group [Formula: see text] where [Formula: see text]. We find the characteristic polynomial of [Formula: see text] for any [Formula: see text] and determine the eigenvalues of [Formula: see text] for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer. We characterize the values of [Formula: see text] for which algebraic and vertex connectivity of [Formula: see text] are equal. We also discuss about the largest and the second largest eigenvalue of [Formula: see text]. Finally, the spectra of [Formula: see text] has been determined for [Formula: see text] where [Formula: see text] are primes and [Formula: see text] is a positive integer.