On Laplacian integrability of comaximal graphs of commutative rings

Abstract

For a commutative ring R, the comaximal graph $\Gamma (R)$ of R is a simple graph with vertex set R and two distinct vertices u and v of $\Gamma (R)$ are adjacent if and only if $aR+bR=R$. In this article, we find the Laplacian eigenvalues of $\Gamma (\mathbb {Z}_{n})$ and show that the algebraic connectivity of $\Gamma (\mathbb {Z}_{n})$ is always an even integer and equals $\phi (n)$, thereby giving a large family of graphs with integral algebraic connectivity. Further, we prove that the second largest Laplacian eigenvalue of $\Gamma (\mathbb {Z}_{n})$ is an integer if and only if $n=p^{\alpha }q^{\beta },$ and hence $\Gamma (\mathbb {Z}_{n})$ is Laplacian integral if and only if $n=p^{\alpha }q^{\beta },$ where p, q are primes and $\alpha , \beta$ are non-negative integers. This answers a problem posed by [Banerjee, Laplacian spectra of comaximal graphs of the ring $\mathbb {Z}_{n}$, Special Matrices, (2022)].