On the nonnegative signed domination number

We investigate the nonnegative signed domination number NNs with emphasis on regular, triangle-free graphs and trees. We give lower and upper bounds on NNs for regular graphs and prove that n=3 is a sharp bound on this parameter for a cubic graph of order n. As an application of Mantel’s theorem we show that NN s (G) can be bounded from below by 2(􀀀2 +p4 + 2n) 􀀀 n, for a triangle-free graph G and characterize all triangle-free graphs for which the equality holds. Also, we bound NN s (T) for a tree T from above and below and characterize all trees attaining the bounds.