Non-Solvable Graphs of Groups

Let G be a group, and $${{\,\mathrm{Sol}\,}}(G)=\{x \in : \langle x,y \rangle \text { is solvable for all } y G\}$$ . We associate graph $$\mathcal {NS}_G$$ (called the non-solvable of G) with whose vertex set $$G \setminus {{\,\mathrm{Sol}\,}}(G)$$ two distinct vertices are adjacent if they generate subgroup. In this paper, we study many properties particular, obtain results on degree, cardinality degree set, realization, domination number, connectivity, independence number clique also consider groups H having isomorphic graphs derive some H. Finally, conclude paper by showing that neither planar, toroidal, double triple toroidal nor projective.