Extremal arborescences in graph orientations

In this paper, we study the extremal number of arborescences over orientations of a fixed graph. We prove that every alternating orientation of a plane Eulerian multigraph minimizes the rooted arborescence count among its Eulerian orientations. We also prove that every tournament of odd order maximizing the total number of arborescences is regular, and hence Eulerian. These results resolve Conjectures 5.1 and 5.4 of Bandekar et al. We further establish upper bounds for arborescence counts in orientations of complete bipartite graphs and characterize the orientations attaining them.