On the maximal energy among orientations of a tree

Abstract EN

The trace norm of a digraph is the trace norm of its adjacency matrix, i.e.

the sum of its singular values.

Given a bipartite graph G, it is well known that the sink-source orientations have minimal trace norm among all orientations of G.

In this paper, we show that the balanced orientations of G attain the maximal trace norm when G is a tree with separated branching vertices, or when G is a double-star tree.

We give examples of trees (with adjacent branching vertices) where non-balanced orientations have maximal trace norm.

This raises the question in general: Which orientations of a tree have maximal trace norm?