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)→{−1,1,2} satisfying the conditions that (i) ∑x∈n(v)f(x)≥0 for each v∈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 ω(f)=∑v∈v(g)f(v). the nonnegative signed total roman domination number γnnstr(g) of g is the minimum weight of an nnstrdf on g. in this paper we initiate the study of the nonnegative signed total roman domination number of graphs, and we present different bounds on γnnstr(g). we determine the nonnegative signed total roman domination number of some classes of graphs. if n is the order and m is the size of the graph g, then we show that γnnstr(g)≥34(8n+1−√+1)−n and γnnstr(g)≥(10n−12m)/5. in addition, if g is a bipartite graph of order n, then we prove that γnnstr(g)≥324n+1√−1)−n.