  An oriented graph $D$ is converse invariant if, for any tournament $T$, the
number of copies of $D$ in $T$ is equal to that of its converse $-D$. El Sahili
and Ghazo Hanna [J. Graph Theory 102 (2023), 684-701] showed that any oriented
graph $D$ with maximum degree at most 2 is converse invariant. They proposed a
question: Can we characterize all converse invariant oriented graphs?
  In this paper, we introduce a digraph polynomial and employ it to give a
necessary condition for an oriented graph to be converse invariant. This
polynomial serves as a cornerstone in proving all the results presented in this
paper. In particular, we characterize all orientations of trees with diameter
at most 3 that are converse invariant. We also show that all orientations of
regular graphs are not converse invariant if $D$ and $-D$ have different degree
sequences. In addition, in contrast to the findings of El Sahili and Ghazo
Hanna, we prove that every connected graph $G$ with maximum degree at least
$3$, admits an orientation $D$ of $G$ such that $D$ is not converse invariant.
We pose one conjecture.
