Some simple groups which are determined by their character degree graphs

Let $G$ be a finite group, and let $\rho(G)$ be the set of prime divisors of the irreducible character degrees of $G$. The character degree graph of $G$, denoted by $\Delta(G)$, is a graph with vertex set $\rho(G)$ and two vertices $a$ and $b$ are adjacent in $\Delta(G)$, if $ab$ divides some irreducible character degree of $G$. In this paper, we are going to show that some simple groups are uniquely determined by their orders and character degree graphs. As a consequence of this paper, we conclude that $M_{12}$ is not determined uniquely by its order and its character degree graph.