Laplacian Spectra of Comaximal Graph of ${\mathbb{Z}}_n$

Abstract

This article focuses on finding the eigenvalues of the Laplacian matrix of the comaximal graph $\Gamma ({\mathbb{Z}}_n)$ of the ring ${\mathbb{Z}}_n$ for $n>2$. We determine the eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ for various $n$ and also provide a procedure to find the eigenvalues of $\Gamma ({\mathbb{Z}}_n)$ for any $n>2$. We show that $\Gamma ({\mathbb{Z}}_n)$ is Laplacian Integral for $n=p^{\alpha }{q}^{\beta }$ where $p,q$ are primes and $\alpha ,\beta$ are non-negative integers. The algebraic and vertex connectivity of $\Gamma ({\mathbb{Z}}_n)$ have been shown to be equal for all $n>2$. An upper bound on the second largest eigenvalue of $\Gamma ({\mathbb{Z}}_n)$ has been obtained and a necessary and sufficient condition for its equality has also been determined. Finally we discuss the multiplicity of the spectral radius and the multiplicity of the algebraic connectivity of $\Gamma ({\mathbb{Z}}_n)$. Some problems have been discussed at the end of this article for further research.