Realization of digraphs in Abelian groups and its consequences

Let $\overrightarrow{G}$ be a directed graph with no component of orderless than~$3$, and let $\Gamma$ finite Abelian group such that $|\Gamma|\geq 4|V(\overrightarrow{G})|$ or if $|V(\overrightarrow{G})|$ is large enough respect to an arbitrarily fixed $\varepsilon>0$ then (1+\varepsilon)|V(\overrightarrow{G})|$. We show there exists injective mapping $\varphi$ from $V(\overrightarrow{G})$ the $\sum_{x\in V(C)}\varphi(x) = 0$ for every connected $C$ $\overrightarrow{G}$, where $0$ identity element $\Gamma$. Moreover we some applications this result distance magic labelings.