  The $\delta$-complement $G_\delta$ of a graph $G$, introduced in 2022 by Pai
et al., is a variant of the graph complement, where two vertices are adjacent
in $G_\delta$ if and only if they are of the same degree but not adjacent in
$G$ or they are of different degrees but adjacent in $G$. In this paper, we
provide the Nordhaus-Gaddum-type bounds, in the spirit of Nordhaus and Gaddum
(1956), over the maximum degrees, the minimum degrees, the vertex
connectivities, and the edge connectivities of a graph and its
$\delta$-complement. All bounds are attained except for the upper bounds on the
product between the minimum degrees of a graph and its $\delta$-complement, the
vertex connectivities of a graph and its $\delta$-complement, and the edge
connectivities of a graph and its $\delta$-complement.
