Oriented bipartite graphs with minimal trace norm. Issue 6 (3rd June 2019)

• Record Type: Journal Article
• Authors: Monsalve, Juan; Rada, Juan
• Is Part Of: Linear & multilinear algebra. Volume 67:Issue 6(2019)
• Page Start: 1121
• Page End: 1131
• Publication Date: 2019-06-03
• Subjects: Trace norm -- orientations of graphs -- digraphs -- bipartite -- unicyclic -- extremal values; 05C50 -- 15A18 -- 05C20; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2018.1448051
• Languages: English
• ISSNs: 0308-1087

Abstract

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, ifΩ 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 inΩ . 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.