On Laplacian integrability of comaximal graphs of commutative rings

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