| Summary: | A signed Italian dominating function on a graph $G=(V,E)$ is a function $f:V\to \{ -1, 1, 2 \}$ satisfying the condition that for every vertex $u$, $f[u]\ge 1$. The weight of signed Italian dominating function is the value $f(V)=\sum_{u\in V}f(u)$. The signed Italian domination number of a graph $G$, denoted by $\gamma_{sI}(G)$, is the minimum weight of a signed Italian dominating function on a graph $G$. In this paper, we determine the signed Italian domination number of some classes of graphs. We also present several lower bounds on the signed Italian domination number of a graph. In particular, for a graph $G$ without isolated vertex we show that $\gamma_{sI}(G)\ge \frac{3n-4m}{2}$ and characterize all graphs attaining equality in this bound. We show that if $G$ is a graph of order $n\ge2$, then $\gamma_{sI}(G)\ge 3\sqrt \frac{n}{2}-n$ and this bound is sharp.
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