Characterization of Finite Groups by Non-Solvable Graphs and Solvabilizers

The non-solvable graph of a finite group $G$‎, ‎denoted by‎ ‎${\cal S}_{G}$‎, ‎is a simple graph whose vertices are the elements‎ ‎of $G$ and there is an edge between two elements $x‎, ‎y\in G$ if and only if‎ ‎$\langle x‎, ‎y\rangle$ is not solvable‎. ‎If $R$ is the solvable‎ ‎radical of $G$‎, ‎the isolated vertices in ${\cal S}_{G}$ are‎ ‎exactly the elements of $R$‎. ‎Thus‎, ‎in the case when $G$ is a‎ ‎non-solvable group‎, ‎it is wise to consider the‎ ‎induced subgraph over $G\setminus R$ which is denoted by‎ ‎$\widehat{{\cal S}_G}$‎. ‎Let $G$ be a finite group and $x\in G$‎. ‎The solvabilizer of $x$ with respect to $G$‎, ‎denoted by‎ ‎$Sol_G(x)$‎, ‎is the set $\{y\in G\ |\ \langle x‎, ‎y\rangle \ {\rm‎ ‎is\ solvable}\}$‎. ‎In this paper‎, ‎we are going to study some properties of‎ ‎$\widehat{{\cal S}_G}$ and the structure of $Sol_G(x)$ for every‎ ‎$x\in G$‎, ‎more precisely‎.