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Let $G$ be a finite group. The prime graph $\Gamma(G)$ of $G$ is defined as follows. The vertices of $\Gamma(G)$ are the primes dividing the order of $G$ and two distinct vertices $p, q$ are joined by an edge if there is an element in $G$ of order $pq$.
It is proved that if $p > 11$ and $p nequiv 1$ (mod 12), then $PSL(2,p)$ is uniquely determined by its prime graph. Also it is proved that if $p > 7$ is a prime number and $\Gamma(G)= \Gamma(PSL(2,p^2))$, then $G \cong PSL(2,p^2)$ or $G \cong PSL(2,p^2)$.2, the non-split extension of $PSL(2,p^2)$ by $\mathbb{Z}_{2}$� In this paper as the main result we determine finite groups $G$ such that $\Gamma(G)= \Gamma(PSL(2,q))$, where $q = p^k$. As a consequence of our results we prove that if $q = p^k$, $k > 1$ is odd and $p$ is an odd prime number, then $PSL(2, q)$ is uniquely determined by its prime graph and so these groups are characterizable by their prime graph.
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