Abstract
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.
Acknowledgements
The author (J.M.) wishes to thank the support of COLCIENCIAS and Becas COLCIENCIAS, Convocatoria No. 727 de 2015, in his process as a Ph.D. Student of Universidad de Antioquia. We are grateful to the anonymous referee for his/her valuable comments which improved this paper.
Disclosure statement
No potential conflict of interest was reported by the authors.