Recognition of the simple groups $PSL_۲(q)$ by character degree graph and order

‎‎Let $G$ be a finite group‎, ‎and $\Irr(G)$ be the set of complex irreducible characters of $G$‎. ‎Let $\rho(G)$ be the set of prime divisors of character degrees of $G$‎. ‎The character degree graph of $G$‎, ‎which is denoted by $\Delta(G)$‎, ‎is a simple graph with vertex set $\rho(G)$‎, ‎and we join two vertices $r$ and $s$ by an edge if there exists a character degree of $G$ divisible by $rs$‎. ‎In this paper‎, ‎we prove that if $G$ is a finite group such that $\Delta(G)=\Delta(\PSL_۲(q))$ and $|G|=|\PSL_۲(q)|$‎, ‎then $G\cong\PSL_۲(q)$‎.

Keywords:

‎character degree graph‎ , ‎simple group‎ , ‎characterization