Laplacian spectrum of weakly zero-divisor graph

Let [Formula: see text] be a commutative ring with unity. The weakly zero-divisor graph [Formula: see text] of [Formula: see text] is the simple undirected graph whose vertices are nonzero zero-divisors of [Formula: see text] and two vertices [Formula: see text], [Formula: see text] are adjacent if and only if there exist [Formula: see text] and [Formula: see text] such that [Formula: see text]. The zero-divisor graph is a spanning subgraph of the weakly zero-divisor graph. Let [Formula: see text] denote the ring of integers modulo [Formula: see text]. It is known that the zero-divisor graph of the ring [Formula: see text], where [Formula: see text] is a prime, is Laplacian integral. In this paper, we obtain the Laplacian spectrum of the weakly zero-divisor graph [Formula: see text] of the ring [Formula: see text] and show that [Formula: see text] is Laplacian integral for arbitrary [Formula: see text]. Consequently, for each [Formula: see text], the weakly zero-divisor graph [Formula: see text] is distance Laplacian integral.