Essential graph of an abelian group

The essential graph of an abelian group $G$, denoted by $\varepsilon(G)$, is an undirected graph with vertex set as the collection of all non-trivial proper subgroups of $G$ and any two distinct vertices $A$ and $B$ are adjacent if and only if $AB$ is an essential subgroup of $G$. We discuss connectedness, traversability and planarity of $\varepsilon(G)$. The concept of a divisor subgroup with respect to a subgroup of $G$ is introduced, whose effect is observed in some of the characterizations of $\varepsilon(G)$.