Group distance magic cubic graphs

A $\mathrm{\Gamma }$\emph{-distance magic labeling} of a graph $G=(V,E)$ with $\mathrm{\mid }V\mathrm{\mid }=n$ is a bijection $\mathrm{\ell }$ from $V$ to an Abelian group $\mathrm{\Gamma }$ of order $n$, for which there exists $\mu \in \mathrm{\Gamma }$, such that the weight $w(x)={\sum }_{y\in N(x)}\mathrm{\ell }(y)$ of every vertex $x\in V$ is equal to $\mu$. In this case, the element $\mu$ is called the \emph{magic constant of} $G$. A graph $G$ is called a \emph{group distance magic} if there exists a $\mathrm{\Gamma }$-distance magic labeling of $G$ for every Abelian group $\mathrm{\Gamma }$ of order $n$. In this paper, we focused on cubic $\mathrm{\Gamma }$-distance magic graphs as well as some properties of such graphs.