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.