Beyond graph energy: norms of graphs and matrices

TL;DR

This paper surveys the study of matrix norms, especially Ky Fan and Schatten norms, as extensions of graph energy, highlighting their combinatorial and analytical connections and open problems.

Contribution

It advances understanding of matrix norms related to graph energy, emphasizing the roles of Hadamard and conference matrices and introducing a new class of matrices.

Findings

Ky Fan and Schatten norms generalize graph energy.

Extremal properties relate to combinatorial structures.

Open problems suggest further research directions.

Abstract

In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.