2-Recognizability by prime graph of $PSL(2,p^2)$

Let $G$ be a finite group and let $\Gamma(G)$ be the prime graph of $G$. Assume $p$ prime. We determine the finite groups $G$ such that $\Gamma(G)=\Gamma(PSL(2,p^2))$ and prove that if $p\ne2,3,7$ is a prime then $k(\Gamma(PSL(2,p^2)))=2$. We infer that if $G$ is a finite group satisfying $|G|=|PSL(2,p^2)|$ and $\Gamma(G)=\Gamma(PSL(2,p^2))$ then $G\cong PSL(2,p^2)$. This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications are also considered of this result to the problem of recognition of finite groups by element orders.