On Laplacian integrability of comaximal graphs of commutative rings

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