  This paper studies intersections of principal blocks of a finite group with
respect to different primes. We first define the block graph of a finite group
$G$, whose vertices are the prime divisors of $|G|$ and there is an edge
between two vertices $p\neq q$ if and only if the principal $p$- and $q$-blocks
of $G$ have a nontrivial common complex irreducible character of $G$. Then we
determine the block graphs of finite simple groups, which turn out to be
complete except those of $J_1$ and $J_4$. Also, we determine exactly when the
Steinberg character of a finite simple group of Lie type lies in a principal
block. Based on the above investigation, we obtain a criterion for the
$p$-solvability of a finite group which in particular leads to an equivalent
condition for the solvability of a finite group. Thus, together with two recent
results of Bessenrodt and Zhang, the nilpotency, $p$-nilpotency and solvability
of a finite group can be characterized by intersections of principal blocks of
some quotient groups.
