The trace norm of r-partite graphs and matrices

TL;DR

This paper investigates the maximum trace norm of r-partite graphs, establishing bounds and identifying cases where these bounds are tight, extending the concept of graph energy to multipartite graphs.

Contribution

It derives new upper bounds for the trace norm of r-partite graphs and identifies conditions under which these bounds are tight, including connections to symmetric conference matrices.

Findings

Established an upper bound for the trace norm of r-partite graphs.

Identified cases where the bound is tight, especially for graphs related to symmetric conference matrices.

Extended the study of graph energy to r-partite graphs with new theoretical insights.

Abstract

The trace norm $\left\Vert G\right\Vert _{\ast}$ of a graph $G$ is the sum of its singular values, i.e., the absolute values of its eigenvalues. The norm $\left\Vert G\right\Vert _{\ast}$ has been intensively studied under the name of graph energy, a concept introduced by Gutman in 1978. This note studies the maximum trace norm of $r$-partite graphs, which raises some unusual problems for $r>2$. It is shown that, if $G$ is an $r$-partite graph of order $n,$ then \[ \left\Vert G\right\Vert _{\ast}<\frac{n^{3/2}}{2}\sqrt{1-1/r}+\left( 1-1/r\right) n. \] For some special $r$ this bound is tight: e.g., if $r$ is the order of a symmetric conference matrix, then, for infinitely many $n,$ there is a graph $G\ $of order $n$ with \[ \left\Vert G\right\Vert _{\ast}>\frac{n^{3/2}}{2}\sqrt{1-1/r}-\left( 1-1/r\right) n.\]