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‎Let $ G $ be a group and let $T^{3}(G)$ be the proper subgroup $\lbrace h\in G \vert (gh)^{3}=(hg)^{3},~for~all‎~ ‎g\in G\rbrace $ of $ G $‎. ‎\textit{The third-noncommuting graph} of $ G $ is the graph with‎ vertex set $ G\setminus T^{3}(G) $‎, ‎where two vertices $ x $ and $ y $ are adjacent if $ (xy)^{3}\neq (yx)^{3} $‎. In this paper‎, ‎at first we obtain some results for this graph for any group $G$‎. ‎Then‎, ‎we investigate the structure of this graph for some groups‎.