  A finite group is called $\psi$-divisible iff $\psi(H)|\psi(G)$ for any
subgroup $H$ of a finite group $G$. Here, $\psi(G)$ is the sum of element
orders of $G$. For now, the only known examples of such groups are the cyclic
ones of square-free order. The existence of non-abelian $\psi$-divisible groups
still constitutes an open question. The aim of this paper is to make a
connection between the $\psi$-divisibility property and graph theory. Hence,
for a finite group $G$, we introduce a simple undirected graph called the
$\psi$-divisibility graph of $G$. We denote it by $\psi_G$. Its vertices are
the non-trivial subgroups of $G$, while two distinct vertices $H$ and $K$ are
adjacent iff $H\subset K$ and $\psi(H)|\psi(K)$ or $K\subset H$ and
$\psi(K)|\psi(H)$. We prove that $G$ is $\psi$-divisible iff $\psi_G$ has a
universal (dominating) vertex. Also, we study various properties of $\psi_G$,
when $G$ is a finite cyclic group. The choice of restricting our study to this
specific class of groups is motivated in the paper.
