On non-cyclic graph of finite groups

Here we study some algebraic properties of non-cyclic graphs. In this paper we show that $\overline{\Gamma}_G$ is isomorphic to $K_3\cup (n-4)K_1$ or $K_4\cup (n-5)K_1$ if and only if $G$ is isomorphic to $D_8$ or $D_{10}$, respectively. We characterize all groups of order $n$ like $G$ in which $\gamma(\Gamma_G)+\gamma(\overline{\Gamma}_G)\in \{n, n-1, n-2, n-3\}$, where $|Cyc(G)|=1$.