Nonnegative signed total Roman domination in graphs

‎Let $G$ be a finite and simple graph with vertex set $V(G)$‎. ‎A nonnegative signed total Roman dominating function (NNSTRDF) on a‎ ‎graph $G$ is a function $f:V(G)rightarrow{-1‎, ‎1‎, ‎2}$ satisfying the conditions‎‎that (i) $sum_{xin N(v)}f(x)ge 0$ for each‎ ‎$vin V(G)$‎, ‎where $N(v)$ is the open neighborhood of $v$‎, ‎and (ii) every vertex $u$ for which‎ ‎$f(u)=-1$ has a neighbor $v$ for which $f(v)=2$‎. ‎The weight of an NNSTRDF $f$ is $omega(f)=sum_{vin V (G)}f(v)$‎. ‎The nonnegative signed total Roman domination number $gamma^{NN}_{stR}(G)$‎ ‎of $G$ is the minimum weight of an NNSTRDF on $G$‎. ‎In this paper we‎‎initiate the study of the nonnegative signed total Roman domination number‎ ‎of graphs‎, ‎and we present different bounds on $gamma^{NN}_{stR}(G)$‎. ‎We determine the nonnegative signed total Roman domination‎‎number of some classes of graphs‎. ‎If $n$ is the order and $m$ the size‎‎of the graph $G$‎, ‎then we show that‎ ‎$gamma^{NN}_{stR}(G)ge frac{3}{4}(sqrt{8n+1}+1)-n$ and $gamma^{NN}_{stR}(G)ge (10n-12m)/5$‎. ‎In addition‎, ‎if $G$ is a bipartite graph of order $n$‎, ‎then we prove‎‎that $gamma^{NN}_{stR}(G)ge frac{3}{2}(sqrt{4n+1}-1)-n$‎.