  The data center networks $D_{n,k}$, proposed in 2008, has many desirable
features such as high network capacity. A kind of generalization of
diagnosability for network $G$ is $g$-good-neighbor diagnosability which is
denoted by $t_g(G)$. Let $\kappa^g(G)$ be the $R^g$-connectivity. Lin et. al.
in [IEEE Trans. on Reliability, 65 (3) (2016) 1248--1262] and Xu et. al in
[Theor. Comput. Sci. 659 (2017) 53--63] gave the same problem independently
that: the relationship between the $R^g$-connectivity $\kappa^g(G)$ and
$t_g(G)$ of a general graph $G$ need to be studied in the future. In this
paper, this open problem is solved for general regular graphs. We firstly
establish the relationship of $\kappa^g(G)$ and $t_g(G)$, and obtain that
$t_g(G)=\kappa^g(G)+g$ under some conditions. Secondly, we obtain the
$g$-good-neighbor diagnosability of $D_{k,n}$ which are
$t_g(D_{k,n})=(g+1)(k-1)+n+g$ for $1\leq g\leq n-1$ under the PMC model and the
MM model, respectively. Further more, we show that $D_{k,n}$ is tightly super
$(n+k-1)$-connected for $n\geq 2$ and $k\geq 2$ and we also prove that the
largest connected component of the survival graph contains almost all of the
remaining vertices in $D_{k,n}$ when $2k+n-2$ vertices removed.
