A note on solvable graphs of finite groups

Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph ${\mathrm{\Gamma }}_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$andtwodistinctvertices$u$and$v$areadjacentifandonlyif$\langle u, v \rangle$issolvable\mathrm{.}Weshowthat$\Gamma_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n\ge 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of ${\mathrm{\Gamma }}_s(G)$ and derive a lower bound of the clique number of ${\mathrm{\Gamma }}_s(G)$. We prove the non-existence of finite non-solvable groups whose solvable graphs are planar, toroidal, double-toroidal, triple-toroidal or projective. We conclude the paper by obtaining a relation between ${\mathrm{\Gamma }}_s(G)$ and the solvability degree of $G$.