In this paper, we will investigate the concept of the torsion-graph of an R-module M, in which the set $T(M)^*$ makes up the vertices of the corresponding torsion graph, ${Gamma}(M)$, with any two distinct vertices forming an edge if $[x:M][y:M]M=0$. We prove that, if ${Gamma}(M)$ contains a cycle, then $gr({Gamma}(M)){leq}4$ and ${Gamma}(M)$ has a connected induced subgraph ${overline{Gamma}}(M)$ with vertex set ${m{in}T(M)^*{mid}Ann(m)M{ eq}0}$ and diam$({overline{Gamma}}(M)){leq}3$. Moreover, if M is a multiplication R-module, then ${overline{Gamma}}(M)$ is a maximal connected subgraph of ${Gamma}(M)$. Also ${overline{Gamma}}(M)$ and ${overline{Gamma}}(S^{-1}M)$ are isomorphic graphs, where $S=R{ackslash}Z(M)$. Furthermore, we show that, if ${overline{Gamma}}(M)$ is uniquely complemented, then $S^{-1}M$ is a von Neumann regular module or ${overline{Gamma}}(M)$ is a star graph.