^{1}Imam Khomeini International University, Qazvin - IRAN.

^{2}Imam Khomeini International University, Qazvin, IRAN.

Abstract

Let $G$ be a group. The order graph of $G$ is the (undirected) graph $\Gamma(G)$, those whose vertices are non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if either $o(H)|o(K)$ or $o(K)|o(H)$. In this paper, we investigate the interplay between the group-theoretic properties of $G$ and the graph-theoretic properties of $\Gamma(G)$. For a finite group $G$, we show that $\Gamma(G)$ is a connected graph with diameter at most two, and $\Gamma(G)$ is a complete graph if and only if $G$ is a $p$-group for some prime number $p$. Furthermore, it is shown that $\Gamma(G)=K_5$ if and only if either $G\cong C_{p^5}, C_3\times C_3$, $C_2\times C_4$ or $G\cong Q_8$.

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