Oriented bipartite graphs with minimal trace norm.

The trace norm of a digraph D is the trace norm of its adjacency matrix A (i.e. the sum of the singular values of A). In particular, when G is a graph then its trace norm is the well-known energy of G introduced by I. Gutman in 1978. In this paper, we address the following problem: among all orientations of a given bipartite graph, which has minimal trace norm? We show that the minimal trace norm is attained in sink-source orientations (i.e. orientations in which every vertex is a sink vertex or a source vertex). Consequently, if is a set of connected bipartite graphs with minimal energy G, then the sink-source orientations of G have minimal trace norm among all orientations of graphs in. This is a consequence of our main result which relates the energy of a graph with the trace norm of its orientations. As another application we find the extremal values of the trace norm among all orientations of unicyclic graphs. [ABSTRACT FROM AUTHOR]