  The aim of this work is to investigate the nonnegative signed domination
number $\gamma^{NN}_s$ with emphasis on regular, ($r+1$)-clique-free graphs and
trees. We give lower and upper bounds on $\gamma^{NN}_s$ for regular graphs and
prove that $n/3$ is the best possible upper bound on this parameter for a cubic
graph of order $n$, specifically. As an application of the classic theorem of
Tur\'{a}n we bound $\gamma^{NN}_s(G)$ from below, for an ($r+1$)-clique-free
graph $G$ and characterize all such graphs for which the equality holds, which
corrects and generalizes a result for bipartite graphs in [Electron. J. Graph
Theory Appl. 4 (2) (2016), 231--237], simultaneously. Also, we bound
$\gamma^{NN}_s(T)$ for a tree $T$ from above and below and characterize all
trees attaining the bounds.
