A graph related to the sum of element orders of a finite group

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

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