كليدواژه :
Prime graph , solvable graph , split graph , finite simple group
چكيده فارسي :
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.
