Order divisor graphs of finite groups

Abstract

For each finite group $G$ we associate a simple undirected graph $O D(G)$, order divisor graph. We investigate the interconnection between the group theoretic properties of $G$ and the graph theoretic properties of order divisor graph $O D(G)$. For a finite group $G$, we obtain the density, the girth and the diameter of $O D(G)$. Further, we obtain the relation $G \cong G^{\prime}$ if and only if $O D(G) \cong O D\left(G^{\prime}\right)$, for every distinct finite groups $G$ and $G^{\prime}$, and $A u t o(G) \subseteq A u t o(O D(G))$.