Essential graph of an abelian group

The essential graph of an abelian group G, denoted by ε(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 ε(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 ε(G).