Extremal norms of graphs and matrices

TL;DR

This paper explores extremal properties of various matrix norms, including Schatten p-norms and Ky Fan k-norms, extending graph energy concepts to broader matrix classes and highlighting differences in Schatten norms across p-values.

Contribution

It generalizes the study of graph energy to Schatten p-norms and Ky Fan k-norms, providing new insights into their extremal behaviors for graphs and matrices.

Findings

Schatten p-norms differ significantly for p<2 and p>=2.

Results extend graph energy concepts to broader matrix norms.

Analysis includes both graphs and general matrices.

Abstract

In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. In this paper some of this research is extended to more general matrix norms, like the Schatten p-norms and the Ky Fan k-norms. Whenever possible the results are given both for graphs and general matrices. In various contexts a puzzling fact was observed: the Schatten p-norms are widely different for 1<=p<2 and for p>=2.