Throughout this paper, every groups are finite. The prime graph of a group $G$ is denotedby $\Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $\Gamma(H) = \Gamma(G)$, we conclude that $G\cong H$. Until now, it is proved that if $k$ is an odd numberand $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ iseven, the recognition by prime graph of $PGL(2,p^k)$, where $p$ is an odd prime number, is anopen problem. In this paper, we generalize this result and we prove that the almost simplegroup $PGL(2,25)$ is recognizable by prime graph. Manuscript profile

The prime graph of a finite group $G$ is denoted by$\Gamma(G)$ whose vertex set is $\pi(G)$ and two distinct primes $p$ and $q$ are adjacent in $\Gamma(G)$, whenever $G$ contains an element with order $pq$. We say that $G$ is unrecognizable by prime graph if there is a finite group $H$ with $\Gamma(H)=\Gamma(G)$, in while $H\not\cong G$. In this paper, we consider finite groups with the same prime graph as the almost simple group $\textrm{PGL}(2,49)$. Moreover, we construct some Frobenius groupswhose prime graphs coincide with $\Gamma(\textrm{PGL}(2,49))$, in particular, we get that $\textrm{PGL}(2,49)$ is unrecognizable by its prime graph. Manuscript profile