Maximum norms of graphs and matrices, and their complements

TL;DR

This paper investigates the maximum trace and Ky Fan norms of graphs and their complements, extending the results to nonnegative matrices and revealing complex optimal structures.

Contribution

It provides the first comprehensive analysis of maximum norms for graphs and their complements, generalizing to nonnegative matrices and exploring Ky Fan norms.

Findings

Maximum trace norms for graphs and complements identified

Analytical matrix functions achieving maxima characterized

Extension to Ky Fan norms and broader matrix classes

Abstract

In this paper, we mainly study the trace norm of the adjacency matrix of a graph, also known as the energy of graph. We give the maximum trace norms for the graph and its complement. 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.