**1. Introduction**

Brauer configuration algebras (BCAs) were introduced recently by Green and Schroll [1]. These algebras are multiserial symmetric algebras whose theory of representation is based on combinatorial data.

Since its introduction, BCAs have been a tool in the research of different fields of mathematics. Its role in algebra, combinatorics, and cryptography is remarkable. For instance, Mali´c and Schroll [2] associated a Brauer configuration algebra to some dessins d'enfants used to study Riemann surfaces, Cañadas et al. investigated the structure of the keys related to the Advanced Encryption Standard (AES) by using some so-called polygonmutations in BCAs. On the other hand, BCAs were a helpful tool for Espinosa et al. to describe the number of perfect matchings in some snake graphs. We point out that Schiffler et al. used perfect matchings of snake graphs to provide a formula for the cluster variables associated with appropriated cluster algebras of surface type. In their doctoral dissertation, Espinosa used the notion of the message of a Brauer configuration to obtain the results [3,4]. According to him, each polygon in a Brauer configuration has associated a word. The concatenation of such words constitutes a message after applying a suitable specialization.

Perhaps, the message associated with a Brauer configuration is one of the most helpful tools to obtain applications of BCAs. In this work, we use Brauer configuration messages, some results of the theory of posets (partially ordered sets) and integer partitions to obtain the trace norm of some {0, 1}-Brauer configurations, which are Brauer configurations whose sets of vertices consist only of 0's and 1's.

It is worth pointing out that the research on trace norm has its roots in chemistry within the Hückel molecular orbital theory (HMO) [5]. Afterwards, Gutman [6] founded an independent line of investigation in spectral graph theory based on graph energy, which is the sum E(*G*) = ∑ *λ*∈*spect*(*MG*) |*λ*|, where *spect*(*MG*) is the set of eigenvalues of

**Citation:** Agudelo Muñetón, N.; Cañadas, A.M.; Espinosa, P.F.F.; Gaviria, I.D.M. {0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory. *Mathematics* **2021**, *9*, 3042. https:// doi.org/10.3390/math9233042

Academic Editors: Irina Cristea and Hashem Bordbar

Received: 26 October 2021 Accepted: 24 November 2021 Published: 26 November 2021

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the adjacency matrix *MG* of a graph *G*. The trace norm associated with the adjacency matrix of a digraph or quiver *Q* denoted ||*Q*||∗ is a generalization of the graph energy. It is also called the Schatten 1-norm, Ky Fan *n*-norm or nuclear norm. If *σ*1, *σ*2, ... , *σ<sup>n</sup>* are the singular values of the *m* × *n*- adjacency matrix *MQ*, with *σ*<sup>1</sup> ≥ *σ*<sup>2</sup> ≥ ··· ≥ *σ<sup>n</sup>* then ||*Q*||∗ <sup>=</sup> min{*m*,*n*}

∑ *i*=1 *σi*. Relationships between energy graph and trace norm were investigated first by Nikiforov [7].

One of the main problems in graph energy theory is giving the extremal values of the energy of significant classes of graphs. For instance, Gutman [6] proved that if *Tn* is a tree with *n* vertices then the following identity holds:

$$\mathcal{E}\left(\mathcal{S}\_n\right) \le \mathcal{E}\left(T\_n\right) \le \mathcal{E}\left(\mathbb{A}\_n\right) \tag{1}$$

where, *Sn* (A*n*) denotes the star (the Dynkin diagram of type A) with *n* vertices.

Graph energy associated with digraphs was investigated first by Kharaghani–Tayfeh– Rezaie [8], afterwards by Agudelo–Nikiforov [9], who found bounds of extremal values of the trace norm for (0, 1)-matrices. It is worth noticing that if the adjacency matrix of a graph *G* is normal, then the graph energy equals the trace norm. In particular, if the adjacency matrix *MG* of a graph *G* is symmetric, then E(*G*) = ||*MG*||∗.

## *Contributions*

In this paper, we introduce the notion of trace norm of a {0, 1}-Brauer configuration. Bounds and explicit values of these trace norms are given for significant classes of graphs induced by this kind of configuration. In particular, the dimension of the associated algebras and their centers are obtained. These results give a relationship between Brauer configuration algebras and graph energy theories with an open problem in the field of integer partitions proposed by Andrews in 1986. Such a problem asks for sets of integer numbers *S*, *T* for which *P*(*S*, *n*) = *P*(*T*, *n* + *a*), where *P*(*X*, *n*) denote the number of integer partitions of *n* into parts within the set *X* with *a* being a fixed positive integer [10].

As a consequence of their investigations regarding Andrews's problem, Cañadas et al. [11,12] introduced and enumerated a particular class of integer compositions (i.e., partitions for which the order of the parts matter) of type D*n*, for which the Andrews's problem holds if *a* = 1. For each *n*, compositions of type D*<sup>n</sup>* constitute a partially ordered set whose number of two-point antichains is given by the integer sequence encoded in the OEIS (On-Line Encyclopedia of Integer Sequences) A344791 [13]. The following identity (2) gives the *n*th term (A344791)*<sup>n</sup>* of this sequence:

$$(\text{A344791})\_n = \sum\_{i=1}^n \sum\_{j=0}^{\lfloor \frac{i}{2} \rfloor} h\_{ij} (t\_i - 2t\_j). \tag{2}$$

where *tk* denotes the *k*th triangular number, and

$$h\_{ij} = \begin{cases} n+1-i, & \text{if } \ i=2j \text{ and } 1 \le j \le \lfloor \frac{n}{2} \rfloor, \\ 0, & \text{if } \ i=n \text{ and } j=0, \\ 1, & \text{otherwise.} \end{cases}$$

This paper uses this sequence to estimate eigenvalues sums of matrices associated with polygons of some {0, 1}-Brauer configurations.

It is worth noting that the relationships introduced in this paper between the theory of Brauer configuration algebras and the graph energy theory do not appear in the current literature devoted to these topics.

This paper is distributed as follows; in Section 2, we recall definitions and notation used throughout the document. In particular, we introduce the notion of trace norm of a {0, 1}-Brauer configuration. In Section 3, we give our main results, we compute and estimate the trace norm and graph energy of some families of graphs defined by Brauer configuration algebras. Concluding remarks are given in Section 4. Examples of trace norm values associated with some Brauer configurations are given in Appendix A.

The following diagram (3) shows how the notions of Brauer configuration and trace norm are related to some of the main results presented in this paper.

#### **2. Background and Related Work**

In this section, we introduce some definitions and notations to be used throughout the paper. In particular, it is given a brief overview regarding the development of the research of graph energy theory, path algebras, and Brauer configuration algebras.

Henceforth, the symbol *A*<sup>∗</sup> will denote the adjoint of a matrix *A*, and *A <sup>F</sup>* the Frobenius norm of a matrix *A*. Furthermore, F is a field, N<sup>+</sup> is the set of positive integers, and *tn* denotes the *n*th triangular number.

#### *2.1. Graph Energy*

The notion of graph energy as the sum of the absolute values of an adjacency matrix was introduced in 1978 by Gutman based on a series of lectures held by them in Stift Rein, Austria [6]. As we explained in the introduction, he was motivated by earlier results regarding the Hückel orbital total *π*-electron energy. According to Gutman and Furtula [14], the results were proposed at that time in good hope that the mathematical community would recognize its significance. However, there was no interest in the subject despite Gutman's efforts, perhaps due to the restrictions imposed on the studied graphs.

The interest in graph energy was renewed at the earliest 2000 when a plethora of results started appearing. Since then, more than one hundred variations of the initial notion have been introduced with applications in different sciences fields. In the same work, Gutman and Furtula claim that an average of two papers per week (more than one hundred in 2017) are written regarding the subject.

Some of the graph energy variations are:


$$(L(G))\_{ij} = \begin{cases} \deg(v\_i) & \text{if } \ i = j, \\ -1 & \text{if } \ i \neq j \text{ and } v\_i \text{ is adjacent to } v\_{j'}, \\ 0 & \text{otherwise.} \end{cases}$$

where deg(*v*) denotes the degree of a vertex *v* in *G*.

3. The *Randi´c energy*, which is the sum of the absolute values of the Randi´c matrix *R*(*G*)=(*R*(*G*)*ij*) of a graph *G*, with

$$(\mathcal{R}(G))\_{ij} = \begin{cases} 0 & \text{if } i = j, \\ \frac{1}{\sqrt{\deg(v\_i)\deg(v\_j)}} & \text{if } v\_i \text{ is adjacent to } v\_j, \\ 0 & \text{otherwise.} \end{cases}$$

Although the notion of graph energy was introduced only for theoretical purposes, currently, its applications embrace a broad class of sciences. The following Table 1 shows some examples of different works devoted to the applications of graph energy and its variations. The authors refer the reader to [14] for more examples of these types of applications.

**Table 1.** Works devoted to the applications of the graph energy theory. In the case of pattern recognition, the applications deal with military purposes.


#### *2.2. Path Algebras*

This section recalls some facts regarding quivers, their associated path algebras, and corresponding module categories. It is worth noting that the quiver or pass graph technique is used in representation theory, and it is an important tool to solve many ring problems, as Belov-Kanel et al. report in [24].

A *quiver Q* = (*Q*0, *Q*1,*s*, *t*) is a quadruple consisting of two sets *Q*<sup>0</sup> whose elements are called *vertices* and *Q*<sup>1</sup> whose elements are called arrows, *s* and *t* are maps *s*, *t* : *Q*<sup>1</sup> → *Q*<sup>0</sup> such that if *α* is an arrow, then *s*(*α*) is called the *source* of *α*, whereas *t*(*α*) is called the *target* of *α* [25]. The adjacency matrix *MQ* and the spectral radius *ρ*(*Q*) = *ρ*(*MQ*) = max|*λ*| (where *λ* runs over all the eigenvalues of *MQ*) of a quiver *Q* are given by those defined by its underlying graph *Q*.

Recall that the adjacency matrix *MG* associated with a graph *G* is defined by the following identities:

$$(M\_G)\_{ij} = \begin{cases} \text{number of edges between } i \text{ and } j, & \text{if } i \neq j, \\ \text{two times the number of loops at } i, & \text{if } i = j. \end{cases}$$

A *path* of length *l* ≥ 1 with source *a* and target *b* is a sequence (*a* | *α*1, *α*2, ... , *α<sup>l</sup>* | *b*) where *t*(*αi*) = *s*(*αi*+1) for any 1 ≤ *i* < *l*. Vertices are paths of length 0 [25–27].

If *Q* is a quiver and F is an algebraically closed field, then the *path algebra* F*Q* of *Q* is the F-algebra whose underlying F-vector space has as basis the set of all paths of length *l* ≥ 0 in *Q*, the natural graph concatenation is the product of two paths [25,26].

An F-algebra Λ is said to be *basic* if it has a complete set {*e*1,*e*2, ... ,*el*} of primitive orthogonal idempotents such that *eiA ejA* for all *i* = *j*.

A *relation* for a quiver *Q* is a linear combination of paths of length ≥ 2 with the same starting points and same endpoints, not all coefficients being zero [25,26].

Let *Q* be a finite and connected quiver. The two-sided ideal of the path algebra F*Q* generated by the arrows of *Q* is called the *arrow ideal* of F*Q* and is denoted by *RQ*, *R<sup>l</sup> <sup>Q</sup>* is the ideal of F*Q* generated as an F-vector space, by the set of all paths of length ≥ *l*. A twosided ideal *I* of the path algebra F*Q* is said to be *admissible* if there exists *m* ≥ 2 such that *R<sup>m</sup> <sup>Q</sup>* ⊆ *<sup>I</sup>* ⊆ *<sup>R</sup>*<sup>2</sup> *Q*.

If *I* is an admissible ideal of F*Q*, the pair (*Q*, *I*) is said to be a *bound quiver*. The quotient algebra F*Q*/*I* is said to be a *bound quiver algebra*.

Gabriel [28] proved that any basic algebra is isomorphic to a bound quiver algebra. He also showed the finiteness criterion for these algebras. Taking into account that one of the main problems in the theory of representation of algebras consists of giving a complete description of the indecomposable modules and irreducible morphisms of the category of finitely generated modules mod Λ of a given algebra Λ.

According to the number of indecomposable modules an algebra Λ can be of finite, tame or wild representation type. We recall that if C is a category of finitely generated modules over an F-algebra Λ (in this case, F is an algebraically closed field). Then a one-parameter family in C is a set of modules of the form:

$$\mathcal{M} = \{ \mathcal{M} / (\mathfrak{x} - a)\mathcal{M} \mid a \in \mathbb{F} \}\tag{4}$$

where M is a Λ − F[*x*]-bimodule, which is finitely generated and free over F[*x*] [29].

Category C is said to be of *tame representation type* or *tame type*, if C = C*n*, and for

*n* every *n*, the indecomposable modules form a *one-parameter* family with maybe finitely many exceptions equivalently in each dimension *d*, all but a finite number of indecomposable Λ-modules of dimension *d* belong to a finite number of one-parameter families. On the other hand, C is of *wild representation type* or *wild type* if it contains *n*-parameter families of indecomposable modules for arbitrarily large *n* [29].

It is worth noting that Drozd in 1977 and Crawley-Boevey in 1988 proved the following result known as the trichotomy theorem.

**Theorem 1** ([30,31], Corollary C)**.** *Let* Λ *be a finite-dimensional algebra over an algebraically closed field. Then* Λ*-mod has either tame type or wild type, and not both.*

The following result proved by Smith establishes a relationship between the theory of representation of algebras and the spectra graph theory.

**Theorem 2** ([32])**.** *Let G be a finite simple graph with the spectral radius (index) ρ*(*G*)*. Then ρ*(*G*) = 2 *if and only if each connected component of G is one of the extended Dynkin diagram* A *n,* D *n,* E 6*,* E 7*,* E 8*. Moreover, <sup>ρ</sup>*(*G*) <sup>&</sup>lt; <sup>2</sup> *if and only if each connected component of <sup>G</sup> is one of Dynkin diagrams* A*n,* D*n,* E6*,* E7*,* E8*.*

**Remark 1.** *Note that if Q is a connected quiver without oriented cycles, then Theorem 2 allows concluding that Q is of finite type (tame type) if and only if ρ*(*Q*) < 2 *(ρ*(*Q*) = 2*). Otherwise, Q is of wild type. A quiver Q has one of these three properties means that the corresponding path algebra* F*Q also does.*

#### *2.3. {0,1}-Brauer Configuration Algebras*

In this section, we discuss some results regarding {0, 1}-Brauer configuration algebras, we refer the reader to [1] for a more detailed study of general Brauer configuration algebras.

{0, 1}-Brauer configuration algebras are bound quiver algebras induced by a Brauer configuration Γ = (Γ0, Γ1, *μ*, O) with the following characteristics:


$$w\_i = w\_{i,1} w\_{i,2} \dots w\_{i, \delta\_i} \dots$$

where *wi*,*<sup>j</sup>* ∈ {0, 1}, *α<sup>i</sup>* = *occ*(0, *Ui*) is the number of times that the vertex 0 occurs in the polygon *Ui*, *δ<sup>i</sup>* − *α<sup>i</sup>* = *occ*(1, *Ui*) is the number of times that the vertex 1 appears in the same polygon with *δ<sup>i</sup>* = |*Ui*| -2.


$$S\_0: \underbrace{\mathcal{U}\_1 < \cdots < \mathcal{U}\_1}\_{a\_1 - \text{times}s} < \underbrace{\mathcal{U}\_2 < \cdots < \mathcal{U}\_2}\_{a\_2 - \text{times}s} < \cdots < \underbrace{\mathcal{U}\_{n-1} < \cdots < \mathcal{U}\_{n-1}}\_{a\_{n-1} - \text{times}s} < \underbrace{\mathcal{U}\_{n-1} < \cdots < \mathcal{U}\_{n-1}}\_{a\_{n-1} - \text{times}s} < \underbrace{\mathcal{U}\_n < \cdots < \mathcal{U}\_n}\_{a\_n - \text{times}s}$$

$$S\_1: \underbrace{\mathcal{U}\_1 < \cdots < \mathcal{U}\_1}\_{(\delta\_1 - a\_1) - \text{times}s} < \underbrace{\mathcal{U}\_2 < \cdots < \mathcal{U}\_2}\_{(\delta\_2 - a\_2) - \text{times}s} < \cdots < \underbrace{\mathcal{U}\_{n-1} < \cdots < \mathcal{U}\_{n-1}}\_{(\delta\_{n-1} - a\_{n-1}) - \text{times}s} < \underbrace{\mathcal{U}\_n < \cdots < \mathcal{U}\_n}\_{(\delta\_n - a\_n) - \text{times}s}$$

Successor sequences is a way of recording how vertices appear in the polygons.

The construction of the quiver *Q*Γ (or simply *Q*, if no confusion arises) goes as follows:


Note that there are different special cycles associated with a vertex *i* ∈ {0, 1} in a polygon *Ui*.

Figure 1 shows the Brauer quiver *Q*<sup>Γ</sup> induced by a {0, 1}-Brauer configuration Γ.

**Figure 1.** Brauer quiver induced by a {0, 1}-Brauer configuration. Symbols *l i j* , *i* ∈ {0, 1}, *j* ∈ {1, 2, . . . , *n*} mean that the corresponding vertex *Uj* has associated *l i <sup>j</sup>* = *occ*(*i*, *Uj*) − 1 different loops.

The *valency val*(*i*) of a vertex *i* ∈ {0, 1} is given by the identity:

$$val(i) = \sum\_{j=1}^{n} occ(i, \mathcal{U}\_j). \tag{5}$$

*val*(*i*) is the number of arrows in the *i*-cycles. A vertex *i* ∈ {0, 1} is said to be *truncated* if *val*(*i*) = 1, otherwise *i* is *non-truncated*. Vertices 0 and 1 are non-truncated in a {0, 1}- Brauer configuration algebra.

The Brauer configuration algebra ΛΓ (or Λ) defined by the quiver *Q* is the path algebra F*Q* bounded by the admissible ideal *I*Γ (or *I*) generated by the following set of relations:


If there exists a word-transformation *T* such that *wi* = *T*(*wi*−1)(*Ri*), for instance, if *wi* = *wi*−1*Ri* with *Ri* a suitable {0,1}-word, then the *cumulative message M*(Γ) of Γ is defined in such a way that *M*(Γ) = *w*1*w*<sup>2</sup> ... *wn* and the *reduced message MR*(Γ) is defined by the concatenation word:

$$M\_R(\Gamma) = w\_1 R\_2 R\_3 \dots R\_n$$

If *MR*(Γ) can be written as a *m* × *n* matrix, then *ρ*(*MR*(Γ)) denotes the *spectral radius of the Brauer configuration* Γ and *the trace norm of the Brauer configuration* Γ is defined as:

$$||M\_R(\Gamma)||\_\* = \sum\_{k=1}^{\min\{m,n\}} \sigma\_k(M\_R(\Gamma)).\tag{6}$$

where *σ*1(*MR*(Γ)) *σ*2(*MR*(Γ)) - ··· *σn*(*MR*(Γ)) - 0 are the singular values of *MR*(Γ), i.e., the square roots of the eigenvalues of *MR*(Γ)*MR*(Γ)∗.

The following Proposition 1 and Theorem 3 prove that the dimension and the center of a Brauer configuration algebra can easily be computed from its Brauer configuration [1,33].

**Proposition 1** ([1], Proposition 3.13)**.** *Let* Λ *be a Brauer configuration algebra associated with the Brauer configuration* Γ *and let* C = {*C*1, ... , *Ct*} *be a full set of equivalence class representatives of special cycles. Assume that for i* = 1, ... , *t, Ci is a special αi-cycle where α<sup>i</sup> is a non-truncated vertex in* Γ*. Then*

$$\dim\_{\mathbb{F}} \Lambda = 2|Q\_0| + \sum\_{\mathbb{C}\_i \in \mathcal{C}} |\mathcal{C}\_i| (n\_i|\mathcal{C}\_i| - 1),$$

*where* |*Q*0| *denotes the number of vertices of Q,* |*Ci*| *denotes the number of arrows in the αi-cycle Ci and ni* = *μ*(*αi*)*.*

**Theorem 3** ([33], Theorem 4.9)**.** *Let* Γ *be a reduced and connected Brauer configuration and let Q be its induced quiver and let* Λ *be the induced Brauer configuration algebra such that* rad2 Λ = 0 *then the dimension of the center of* Λ *denoted* dim<sup>F</sup> *Z*(Λ) *is given by the formula:*

$$\dim\_{\mathbb{F}} Z(\Lambda) = 1 + \sum\_{\mathfrak{a} \in \Gamma\_0} \mu(\mathfrak{a}) + |\Gamma\_1| - |\Gamma\_0| + \mathfrak{\*} (Loop \mathbb{Q}) - |\mathbb{C}\_\Gamma|.$$

*where* |CΓ| = {*α* ∈ Γ<sup>0</sup> | *val*(*α*) = 1, *and μ*(*α*) > 1}*.*

In this case, rad *M* denotes the radical of a module *M*, rad *M* is the intersection of all the maximal submodules of *M*.

The following are properties of {0, 1}-Brauer configuration algebras based on Proposition 1 and Theorem 3.

**Corollary 1.** *Let* Λ *be a Brauer configuration algebra induced by a {0,1}-Brauer configuration* Γ = (Γ0, Γ1, *μ*, O) *with rad*<sup>2</sup> Λ = 0*. Then the following statements hold:*

