On non-cyclic graph of finite groups

Here we study some algebraic properties of non-cyclic graphs. In this paper we show that ${\bar{\mathrm{\Gamma }}}_G$ 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 $\gamma ({\mathrm{\Gamma }}_G)+\gamma ({\bar{\mathrm{\Gamma }}}_G)\in \{n,n-1,n-2,n-3\}$, where $\mathrm{\mid }Cyc(G)\mathrm{\mid }=1$.