On the sum of k largest singular values of graphs and matrices

TL;DR

This paper investigates the sum of the k largest singular values of graphs and matrices, exploring their properties and relationships with key graph parameters, extending results to broader matrix classes.

Contribution

It provides new insights into Ky Fan k-norms of graphs and matrices, connecting them with spectral and combinatorial graph parameters, and extends existing results to more general matrices.

Findings

Relations between Ky Fan k-norms and chromatic number

Connections to spectral radius and spread

Extensions to broader classes of matrices

Abstract

In the recent years, the trace norm of graphs has been extensively studied under the name of graph energy. The trace norm is just one of the Ky Fan k-norms, given by the sum of the k largest singular values, which are studied more generally in the present paper. Several relations to chromatic number, spectral radius, spread, and to other fundamental parameters are outlined. Some results are extended to more general matrices.