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Let $\textit{G}$ be a finite group. We define the prime graph
$\Gamma{(G)}$ as follows. The vertices of $\Gamma{(G)}$ are the
primes dividing the order of $\textit{G}$ and two distinct
vertices $\textit{p, q}$ are joined by an edge if there is an
element in $\textit{G}$ of order $\textit{pq}$. Recently M. Hagie
in (Hagie, M. (2003), The prime graph of a sporadic simple group,
Comm. Algebra, 31: 4405-4424) determined finite groups
$\textit{G}$ satisfying $\Gamma(G)=\Gamma(S)$, where $\textit{S}$
is a sporadic simple group. Let $\textit{p}>3$ be a prime number.
In this paper we determine finite groups $\textit{G}$ such that
$\Gamma(G)=\Gamma(PSL(2,p))$. ALso is prove that if
$\textit{p}>11$ and $p\not\equiv 1$ (mod 12), then
$\textit{PSL}(2,p)$ is uniquely determined by its prime graph. As
a consequence of our results we can give positive answer to a
conjecture of W.Shi and J. Bi for the group $\textit{PSL}(2,p)$.
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