  Foata and Zeilberger defined the graphical major index, $\mathrm{maj}'_U$,
and the graphical inversion index, $\mathrm{inv}'_U$, for words. These
statistics are a generalization of the classical permutation statistics
$\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed
that $\mathrm{maj}'_U$ and $\mathrm{inv}'_U$ are equidistributed over all
rearrangement classes if and only if $U$ is bipartitional. In this paper we
strengthen their result by showing that if $\mathrm{maj}'_U$ and
$\mathrm{inv}'_U$ are equidistributed on a single rearrangement class then $U$
is essentially bipartitional. Moreover, we define a graphical sorting index,
$\mathrm{sor}'_U$, which generalizes the sorting index of a permutation. We
then characterize the graphs $U$ for which $\mathrm{sor}'_U$ is equidistributed
with $\mathrm{inv}'_U$ and $\mathrm{maj}'_U$ on a single rearrangement class.
