Almost Simple Groups and Their Non- Commuting Graph

‎Let $G$ be a non-abelian finite group and $Z(G)$ be the center of‎ ‎$G$‎. ‎The non-commuting graph‎, ‎$\nabla(G)$ associated to $G$ is the‎ ‎graph whose vertex set is $G-Z(G)$ and two distinct vertices $x‎, ‎y$‎ ‎are adjacent if and only if $xy\neq yx$‎. ‎We conjecture that if $G$‎ ‎is an almost simple group and $H$ is a non-abelian finite group such‎ ‎that $\nabla(G)\cong \nabla(H)$‎, ‎then $|G|=|H|$‎. ‎Among other‎ ‎results‎, ‎we prove that if $(G:S)$ is an almost simple group such‎ ‎that $S$ is one of the Sporadic simple groups or $S$ is one of the‎ ‎mentioned Lie Groups $L_2(7)‎, ‎L_2(8)‎, ‎L_2(17)‎, ‎L_3(3)‎, ‎U_3(3)‎, ‎U_4(2)‎, ‎F_4(2)‎, ‎O^+_{10}(2)$ and $O^-_{10}(2)$ and $H$ is a‎ ‎non-abelian group with isomorphic non-commuting graphs‎, ‎then $G\cong‎ ‎H$