Non-cyclic graph associated with a group

We associate a graph ${\mathcal{C}}_G$ to a non locally cyclic group $G$ (called the non-cyclic graph of $G$) as follows: take $G\mathrm{\backslash }Cyc(G)$ as vertex set, where $Cyc(G)=\{x\in G\mathrm{\mid }<x,y>\text{is cyclic for all}y\in G\}$ is called the cyclicizer of $G$, and join two vertices if they do not generate a cyclic subgroup. For a simple graph $\mathrm{\Gamma }$, $w(\mathrm{\Gamma })$ denotes the clique number of $\mathrm{\Gamma }$, which is the maximum size (if it exists) of a complete subgraph of $\mathrm{\Gamma }$. In this paper we characterize groups whose non-cyclic graphs have clique numbers at most 4. We prove that a non-cyclic group $G$ is solvable whenever $w({\mathcal{C}}_G)<31$ and the equality for a non-solvable group $G$ holds if and only if $G\mathrm{/}Cyc(G)\cong A_5$ or $S_5$.