The trace norm of r-partite graphs and matrices

The trace norm ‖G‖∗ of a graph G is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm ‖G‖∗ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note focuses on the maximum trace norm of r-partite graphs, which for r > 2 raises some unusual problems. It is shown that, if G is an r-partite graph of order n, then ‖G‖∗ < n3/2 2 √ 1− 1/r + (1− 1/r)n. For some special r this bound is tight: e.g., if r is the order of a real symmetric conference matrix, then, for inifinitely many n, there is a graph G of order n with ‖G‖∗ > n3/2 2 √ 1− 1/r − (1− 1/r)n. AMS classification: 15A42; 05C50.