Maximum norms of graphs and their complements

Given a graph G; let kGk denote the trace norm of its adjacency matrix, also known as the energy of G: The main result of this paper states that if G is a graph of order n; then kGk + G (n 1) 1 +n ; where G is the complement of G: Equality is possible if and only if G is a strongly regular graph with parameters (n; (n 1) =2; (n 5) =4; (n 1) =4) ; known also as a conference graph. In fact, the above problem is stated and solved in a more general setup for nonnegative matrices with bounded entries. In particular, this study exhibits analytical matrix functions attaining maxima on matrices with rigid and complex combinatorial structure. In the last section the same questions are studied for Ky Fan norms. Possibe directions for further research are outlined, as it turns out that the above problems are just a tip of a larger multidimensional research area. AMS classi…cation: 15A42, 05C50 Keywords: graph eigenvalues; complementary graph; trace norm; graph energy; Nordhaus-Gaddum problem. 1 Introduction and main results In this paper we study the maxima of certain norms of nonnegative matrices with bounded entries. We shall focus …rst on the trace norm kAk of a matrix A; which is just the sum of its singular values. The trace norm of the adjacency matrix of graphs has been intensively studied recently under the name graph energy. This research has been initiated by Gutman in [3]; the reader is referred to [6] for a comprehensive recent survey and references. One of the most intriguing problems in this area is to determine which graphs with given number of vertices have maximal energy. A cornerstone result of Koolen and Moulton [5] shows that if G is a graph of order n; then the trace norm of its adjacency matrix