On the maximal energy among orientations of a tree
Keywords:
Trace norm, digraph, orientation of a tree, extremal values.Abstract
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 and raise the question in general: which
orientations of a tree have maximal trace norm?
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