Hall Graph of a Finite Group

The Hall graph of a finite group $G$ is a simple graph whose vertex set is $\pi(G)$, the set of all prime divisors of its order, and two distinct primes $p$ and $q$ are joined by an edge if $G$ has at least one Hall $\{p, q\}$-subgroup. For all primes $p_1 \cdots p_k$ of $\pi(G)$, we call the $k$-tuple ${\rm D}_{\rm H}(G)=(d_{\rm H}(p_1), \ldots, d_{\rm H}(p_k))$, the degree pattern of Hall graph of $G$, where $d_{\rm H}(p)$ signifies the degree of vertex $p$. This paper provides some properties of Hall graph. It also gives a characterization for some finite simple groups via order and degree pattern of Hall graph.