  The interior polynomial of a directed graph is defined as the
$h^*$-polynomial of the graph's (extended) root polytope, and it displays
several attractive properties. Here we express its degree in terms of the
minimum cardinality of a directed join, and give a formula for the leading
coefficient. We present natural generalizations of these results to oriented
regular matroids; in the process we also give a facet description for the
extended root polytope of a regular oriented matroid.
  By duality, our expression for the degree of the interior polynomial implies
a formula for the degree of the parking function enumerator of an Eulerian
directed graph (which is equivalent to the greedoid polynomial of the
corresponding branching greedoid). We extend that result to obtain the degree
of the parking function enumerator of an arbitrary rooted directed graph in
terms of the minimum cardinality of a certain type of feedback arc set.
