Split Prime and Solvable Graphs

‎The prime (resp‎. ‎solvable) graph ${\rm GK}(G)$ (resp‎. ‎$\mathcal{S}(G)$) of a finite group‎ ‎$G$ is a simple graph whose vertices are the prime divisors of‎ ‎the order of $G$ and two distinct vertices $p$ and $q$ are joined‎ ‎by an edge if and only if $G$ has a cyclic (resp‎. ‎solvable) subgroup of order‎ ‎divisible by $pq$‎. ‎In this talk‎, ‎we first show that the prime graph‎ ‎of any alternating and sporadic simple groups is split‎, ‎that is‎, ‎a graph whose‎ ‎vertex set can be partitioned into two sets such that the induced‎ ‎subgraph on one of them is a complete graph and the induced‎ ‎subgraph on the other is an independent set‎. ‎Next‎, ‎we prove that‎ ‎the solvable graph of any alternating and sporadic simple groups‎ ‎is split‎, ‎except for the following simple groups‎: ‎$M_{22}$‎, ‎$M_{23}$‎, ‎$M_{24}$‎, ‎$Co_3$‎, ‎$Co_2$‎, ‎$Fi_{23}$‎, ‎$Fi_{24} $‎, ‎$B$‎, ‎$M$ and $J_4$‎. ‎Finally‎, ‎we consider the compact‎ ‎form of a prime graph and show that the‎ ‎compact form of a nonabelian simple group is split‎.