Abstract: In this paper, we determine the Laplacian spectrum of the nil-clean graph $G_{NC}(\mathbb{Z}_{n})$ for the ring $\mathbb{Z}_n$, obtain a necessary and sufficient condition for the Laplacian spectral radius of $G_{NC}(\mathbb{Z}_{n})$ to be equal to $n$, and furthermore, address the problem of the coincidence of the algebraic connectivity and the vertex connectivity of $G_{NC}(\mathbb{Z}_{n})$.

Key words: Nil-clean graph, Laplacian spectrum, Laplacian spectral radius, Algebraic connectivity, Vertex connectivity