Let R be a commutative ring with identity $1\ne 0$ and let $Z{\left(R\right)}^{\prime }$ be the set of all non-unit and non-zero elements of ring R. ${\Gamma }^{\prime }\left(R\right)$ denotes the cozero-divisor graph of R and is an undirected graph with vertex set $Z{\left(R\right)}^{\prime }$, $w\notin zR$, and $z\notin wR$ if and only if two distinct vertices w and z are adjacent, where $qR$ is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs ${\Gamma }^{\prime }\left({\mathbb{Z}}_n\right)$. We also show that the cozero-divisor graph ${\Gamma }^{\prime }\left({\mathbb{Z}}_{p_1{p}_2}\right)$ is a signless Laplacian integral.

Let 𝑅 be a commutative ring with identity 1 ≠ 0 and let Z ⁢ ( R ) ′ be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ , and w ∉ z ⁢ R and z ∉ w ⁢ R if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) for n = p 1 N ⁢ p 2 ⁢ p 3 and p 1 N ⁢ p 2 M ⁢ p 3 , where p 1 , p 2 , p 3 are distinct primes and N , M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) is a signless Laplacian integral.

Let $\mathcal{A}$ be a unital ∗-algebra over the complex fields $\mathbb{C}$. In this article, it is proved that a nonlinear mixed bi-skew Jordan n-derivation is an additive ∗-derivation under certain conditions. As applications, the main result is applied to some special classes of ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra and von Neumann algebra with no central summands of type $I_1$.