The energy $\mathcal{E}\left(G\right)$ of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a $\{0,1\}$-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

The four subspace problem is a known matrix problem, which is equivalent to determining all the indecomposable representations of a poset consisting of four incomparable points. In this paper, we use solutions of this problem and invariants associated with indecomposable projective modules with some suitable Brauer configuration algebras to categorify the integer sequence encoded in the OEIS as A100705 and some related integer sequences.

The trace norm of a matrix is the sum of its singular values. This paper presents results on the minimum trace norm $\psi_{n}\left( m\right) $ of $\left( 0,1\right) $-matrices of size $n\times n$ with exactly $m$ ones. It is shown that: (1) if $n\geq2$ and $n<m\leq2n,$ then $\psi_{n}\left( m\right) \leq \sqrt{m+\sqrt{2\left( m-1\right) }}$ , with equality if and only if $m$ is a prime; (2) if $n\geq4$ and $2n<m\leq3n,$ then $\psi_{n}\left( m\right) \leq \sqrt{m+2\sqrt{2\left\lfloor m/3\right\rfloor }}$ , with equality if and only if $m$ is a prime or a double of a prime; (3) if $3n<m\leq4n,$ then $\psi_{n}\left( m\right) \leq\sqrt{m+2\sqrt{m-2}}% $ , with equality if and only if there is an integer $k\geq1$ such that $m=12k\pm2$ and $4k\pm1,6k\pm1,12k\pm1$ are primes.