*2.4. Posets*

A partially ordered set (or *poset*) is an ordered pair (P, ≤) where P is a not empty set, and ≤ is a partial order over the elements of P, i.e., ≤ is reflexive, antisymmetric, and transitive. Henceforth, if no confusion arises we will write P instead of (P, ≤) to denote a partially ordered set.

For each *x*, *y* ∈ P, if *x* ≤ *y* or *y* ≤ *x*, we say that *x* and *y* are *comparable points*, whereas if *x y* and *y x*, we say that *x* and *y* are *incomparable points* (the subset {*x*, *y*} is a *two-point antichain*), this situation is denoted by *x y*. An ordered set C is called a *chain* (or a totally ordered set or a linearly ordered set) if and only if for all *x*, *y* ∈ C we have *x* ≤ *y* or *y* ≤ *x* (i.e., *x* and *y* are comparable points).

A relation *x* ≤ *y* in a poset P is said to be a covering, if for any *z* ∈ P such that *x* ≤ *z* ≤ *y* it holds that *x* = *z* or *y* = *z* [34].

#### **3. Applications**

In this section, we give applications of {0, 1}-Brauer configuration algebras in graph energy. We start by defining some suitable {0, 1}- Brauer configuration algebras, dimensions of these algebras and corresponding centers are given as well. We also compute and estimate eigenvalues and trace norm of their reduced messages *MR*(Γ).

1. For *n* - 2 fixed, let us consider the {0,1}-Brauer configuration Δ*<sup>n</sup>* = (Δ*<sup>n</sup>* <sup>0</sup> , <sup>Δ</sup>*<sup>n</sup>* <sup>1</sup> , *μ*, O), such that:

$$\begin{aligned} \Lambda\_0^\pi &= \{0, 1\}. \\ \Lambda\_1^\pi &= \{D\_1, D\_2, \dots, D\_\pi\}, \quad \text{for } 1 \le i \le n, \quad |D\_i| = (t\_{i+2} - 1)^2. \\ \mu(0) &= \mu(1) = 1. \end{aligned} \tag{7}$$

The orientation O is defined in such a way that in successor sequences associated with vertices 0 and 1, it holds that *Di* < *Di*+1, for 1 *i n*.

Polygons *Di* can be seen as (*ti*+<sup>2</sup> − 1) × (*ti*+<sup>2</sup> − 1)-matrices over Z<sup>2</sup> or as (*ti*+<sup>2</sup> − 1) × 1-matrices over the vector space *Pti*<sup>+</sup>2−<sup>2</sup> of polynomials of degree ≤ *ti*+<sup>2</sup> − 2. Its construction goes as follows:

(a) For any *i*, 1 ≤ *i* ≤ *n*, *Di* is a symmetric matrix,

$$\begin{aligned} \text{(b)} \qquad D\_{1} &= \begin{bmatrix} 1 & 1 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} t^{4} + t^{3} + t + 1 \\ t^{4} + t^{3} + t^{2} + 1 \\ t^{3} + t^{2} + t + 1 \\ t^{4} + t^{2} + t + 1 \\ t^{4} + t^{3} + t^{2} + t + 1 \end{bmatrix}, \\ \text{(c)} \qquad D\_{i} &= \begin{bmatrix} \underbrace{B\_{i-1}^{i+1}}\_{\begin{bmatrix} B\_{i}^{i+1} \\ B\_{i}^{i+1} \end{bmatrix}}\_{\begin{bmatrix} B\_{i}^{i+1} \\ B\_{i+1}^{i+1} \end{bmatrix}} \\ \text{(d)} \qquad \text{Block } & \underset{\begin{bmatrix} B\_{i}^{i+1} \\ B\_{i+1}^{i+1} \end{bmatrix}}{\begin{bmatrix} B\_{i+1}^{i+1} \\ B\_{i+1}^{i+1} \end{bmatrix}} \end{aligned}$$

(d) Blocks *Bi*+*<sup>k</sup> <sup>j</sup>* , with *k* > 1 are defined as follows:

$$\text{i.} \qquad \text{Over } \mathbb{Z}\_{2^{\prime}} \quad \mathbb{B}\_{j}^{j} \in M\_{(j+1)\times(j+1)^{\prime}} \quad \mathbb{B}\_{j}^{j+s} \in M\_{(j+1)\times(j+s+1)^{\prime}} \ 0 \le s \le j+1, j \in \mathbb{N}$$

$$\begin{array}{ll} \text{ii.} & \text{Over } P\_{j+s+2} & \mathbb{B}\_{j}^{i+k} = \begin{bmatrix} p\_{j}^{i+k}(t) \\ p\_{j}^{i+k}(t) \\ \vdots \\ p\_{j+1}^{i+k}(t) \end{bmatrix}, \\\\ p\_{h}^{j+k}(t) &= \begin{cases} \sum\_{l=0}^{j-h+1} x^{l} & \text{if } 1 \le h \le k, \\\ \sum\_{l=0}^{j-h+1} x^{l} + \sum\_{j=0}^{h-k-1} x^{j+k-h+1}, & \text{if } h > k \text{ and } 2 \le k \le i+2, \\\ p\_{h}^{m}(t), & \text{if } m > i+2. \end{cases} \end{array}$$

**Corollary 2.** *If* D*<sup>n</sup>* = F*Q<sup>n</sup>* Δ*n*/*I<sup>n</sup>* <sup>Δ</sup>*<sup>n</sup> is the Brauer configuration algebra induced by the {0,1}- Brauer configuration* Δ*<sup>n</sup> then the following statements hold:*

$$\begin{aligned} \dim\_{\mathbb{F}} \mathfrak{D}^{\mathfrak{n}} &= \left(\mathfrak{e}\_{n} - d\_{\mathfrak{n}}\right)^{2} + \left(\mathfrak{e}\_{n} - 1\right)^{2} + \left(d\_{\mathfrak{n}} - 1\right), \\ \dim\_{\mathbb{F}} Z(\mathfrak{D}^{\mathfrak{n}}) &= \left(t\_{n+2}\right)^{2} + n + 3. \end{aligned} \tag{8}$$

*where*

$$\begin{aligned} a\_n &= \frac{1 - (-1)^n - 8n - 4n^2 + 8n^3 + 2n^4}{32} = (\text{A344791})\_n \\ b\_{n+2} &= \sum\_{i=1}^{n+2} t\_i^2 - 10, \\ c\_{n+2} &= -\frac{(n+2)(n+3)(n+4)}{3} + 8, \\ d\_n &= b\_{n+2} + c\_{n+2} + n, \quad n \ge 1, \\ e\_n &= 2 \sum\_{i=1}^n a\_{i+1}. \end{aligned} \tag{9}$$

**Proof.** For *n* > 1 fixed, consider the following set:

$$\mathcal{P}\_n = \{ \mathbf{x}\_{1,1}, \mathbf{x}\_{1,2}, \mathbf{x}\_{2,1}, \mathbf{x}\_{2,2}, \mathbf{x}\_{2,3}, \dots, \mathbf{x}\_{i,1}, \dots, \mathbf{x}\_{i,l+1}, \dots, \mathbf{x}\_{n,1}, \dots, \mathbf{x}\_{n,n+1} \} \tag{10}$$

P*<sup>n</sup>* is endowed with a partial order , which defines the following coverings:

$$\begin{aligned} \chi\_{j,k} &\lhd \chi\_{j,k+1}, \quad 1 \le j \le n, \quad 1 \le k \le j, \\ \chi\_{j,k} &\upharpoonright \chi\_{j+1,k+1}, \quad 1 \le j < n, \quad 1 \le k \le j+1, \\ \chi\_{r,k} &\upharpoonright \chi\_{r-1,k+1}, \quad 1 < r \le n, \quad 1 \le k \le r. \end{aligned} \tag{11}$$

(P*n*, ) defines a matrix *Mn* whose entries *mi*,*<sup>j</sup>* are given by the following identities:

$$m\_{i,j} = \begin{cases} 1, & \text{if } \ x\_{i,r} \preceqcurlyeq x\_{j,s} \quad \text{or} \quad x\_{j,s} \preceqcurlyeq x\_{i,r}, \\ 0, & \text{otherwise}. \end{cases}$$

Clearly *Mn* is a (*tn*+<sup>1</sup> − 1) × (*tn*+<sup>1</sup> − 1) symmetric matrix with *Mn* = *Dn*−<sup>1</sup> ∈ Δ*<sup>n</sup>* <sup>1</sup> , that is, *Mn* is the matrix associated with the polygon *Dn*−<sup>1</sup> ∈ Δ*<sup>n</sup>* <sup>1</sup> . Thus, <sup>1</sup> <sup>2</sup>occ(0, *Dn*) equals the number of two-point antichains in (P*n*, ). Therefore, occ(0, *Dn*) is twice the *n*th term of the sequence A344791 (see (2), (9)), and occ(1, *Dn*)=(*tn*+<sup>1</sup> − 1)<sup>2</sup> − occ(0, *Dn*). Since dim<sup>F</sup> D*<sup>n</sup>* = 2*n* + *val*(0)(*val*(0) − 1) + *val*(1)(*val*(1) − 1). The result holds. Since rad<sup>2</sup> D*<sup>n</sup>* = 0, then dim<sup>F</sup> *Z*(D*n*) = 1 + *n* + #(*Loops Q*Δ*<sup>n</sup>* ) with #(*Loops Q*Δ*<sup>n</sup>* ) = (*tn*+2)<sup>2</sup> + 2. We are done.

Now we are interested in estimating the eigenvalues of *Mn*. Since the polygons *Dn* ∈ Δ*<sup>n</sup>* <sup>1</sup> can be seen as (*tn*+<sup>1</sup> − 1) square symmetric matrices described in the previous proof as *Dn*−<sup>1</sup> = *Mn*. We will assume that for each *n*, the real eigenvalues of a matrix *Mn* are indexed in the following decreasing order:

$$
\mu\_{\max}(M\_{\mathfrak{n}}) = \mu\_1(M\_{\mathfrak{n}}) \succeq \mu\_2(M\_{\mathfrak{n}}) \succeq \dots \succeq \mu\_{\mathfrak{l}\_{\mathfrak{n}+1}-1}(M\_{\mathfrak{n}}) = \mu\_{\min}(M\_{\mathfrak{n}}) \square
$$

The next result, which derives two inequalities for the eigenvalues of Hermitian matrices, was proved by Bollobás and Nikiforov [35].

**Theorem 4** ([35], Theorem 2)**.** *Suppose that* 2 *k n and let A* = (*aij*) *be a Hermitian matrix of size n. For every partition* {1, 2, . . . , *n*} = *N*<sup>1</sup> ∪···∪ *Nk we have*

$$
\mu\_1(A) + \dots + \mu\_k(A) \geqslant \sum\_{r=1}^k \frac{1}{|N\_r|} \sum\_{i,j \in N\_r} a\_{ij}
$$

*and*

$$
\mu\_{k+1}(A) + \dots + \mu\_n(A) \lesssim \sum\_{r=1}^k \frac{1}{|N\_r|} \sum\_{i,j \in N\_r} a\_{ij} - \frac{1}{n} \sum\_{i,j \in \{1,2,\dots,n\}} a\_{ij} \dots$$

The following result on the eigenvalues of *Mn* can be obtained by applying Theorem 4 to the matrix *Mn* associated with the polygon *Dn*−<sup>1</sup> ∈ Δ*<sup>n</sup>* 1 .

**Corollary 3.** *For n* > 1 *and k* = *n. Let Mn* = (*mij*) *be the matrix associated with the polygon Dn*−<sup>1</sup> ∈ Δ*<sup>n</sup>* <sup>1</sup> *. For partition* {1, 2, ... , *tn*+<sup>1</sup> <sup>−</sup> <sup>1</sup>} <sup>=</sup> *<sup>N</sup>*<sup>1</sup> ∪···∪ *Nn where Ni* <sup>=</sup> *i*(*i*+1) <sup>2</sup> ,..., *<sup>i</sup>*(*i*+3) 2 *. We have*

$$\sum\_{i=1}^{n} \mu\_i(M\_n) \gtrless t\_{n+1} - 1 \tag{12}$$

and 
$$\sum\_{i=n+1}^{t\_{n+1}-1} \mu\_i(M\_n) \le \frac{2(\text{A344791})\_n}{t\_{n+1}-1}. \text{ (see (2)).}\tag{13}$$

**Proof.** Since *Ni* = *i*(*i*+1) <sup>2</sup> ,..., *<sup>i</sup>*(*i*+3) 2 , for each *i* = {1, 2, ... , *n*} then |*Ni*| = *i* + 1, besides each set *Ni* can be seen as a subset of the set P*<sup>n</sup>* defined in (10) as follows:

$$N\_i = \{ \mathfrak{x}\_{i,1}, \dots, \mathfrak{x}\_{i,i+1} \}.$$

On the other hand, to compute ∑*i*,*j*∈*Ni aij*, we will use the coverings defined in (11) and the fact that P*<sup>n</sup>* is a partial order, so we obtain:

$$\begin{aligned} \sum\_{i,j \in N\_i} m\_{ij} &= 2 \sum\_{j=1}^{i} (\mathbf{x}\_{i,j} \preceq \mathbf{x}\_{i,j+1}) + \sum\_{j=1}^{i+1} (\mathbf{x}\_{i,j} \preceq \mathbf{x}\_{i,j}) + 2 \sum\_{j=1}^{i-1} (\mathbf{x}\_{i,j} \preceq \mathbf{x}\_{i,j+2}) \\ &= 2i + (i+1) + 2t\_{i-1} \\ &= (i+1)^2 \end{aligned}$$

Therefore:

$$\begin{array}{rcl} \sum\_{i=1}^{n} \frac{1}{|N\_i|} \sum\_{i,j \in N\_i} m\_{ij} & = i+1 \text{ and} \\ \frac{1}{t\_{n+1}-1} \sum\_{i,j \in \{1,2,\dots,t\_{n+1}-1\}} m\_{i,j} & = \frac{1}{t\_{n+1}-1} ||M\_n||\_F^2 = \frac{1}{t\_{n+1}-1} \left( (t\_{n+1}-1)^2 - 2(\text{A344791})\_n \right) \end{array}$$

Hence, applying Theorem 4 we obtain (12) and (13).

2. For *n* - 1 fixed, let Γ*<sup>n</sup>* = {Γ*<sup>n</sup>* <sup>0</sup> , <sup>Γ</sup>*<sup>n</sup>* <sup>1</sup> , *μ*, O} be a {0,1}-Brauer configuration such that:

$$\begin{aligned} \Gamma\_0^n &= \{0, 1\}. \\ \Gamma\_1^n &= \{\mathcal{U}\_1, \mathcal{U}\_2, \dots, \mathcal{U}\_n\}, \quad \text{for } 1 \le i \le n, \quad |\mathcal{U}\_i| = 2^{2n}. \\ \mu(0) &= \mu(1) = 1. \end{aligned} \tag{14}$$

The orientation O is defined in such a way that in successor sequences associated with vertices 0 and 1, it holds that *Ui* < *Ui*+1.

Polygons *Ui* can be seen as 2*<sup>n</sup>* × 2*n*-matrices over Z<sup>2</sup> using the Kronecker product, denoted by ⊗, as follows:

$$\begin{aligned} \mathcal{U}\_1 &= \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \\ \mathcal{U}\_2 &= \mathcal{U}\_1 \otimes \mathcal{U}\_1 \\ &\vdots \\ \mathcal{U}\_i &= \mathcal{U}\_1 \otimes \mathcal{U}\_{i-1} \end{aligned} \tag{15}$$

**Corollary 4.** *For n* - 1*, if* G*<sup>n</sup>* = F*Q<sup>n</sup>* Γ*n*/*I<sup>n</sup>* <sup>Γ</sup>*<sup>n</sup> is the Brauer configuration algebra induced by the {0,1}-Brauer configuration* Γ*<sup>n</sup> then the following statements hold:*

$$\dim\_{\mathbb{F}} \mathfrak{G}^n = 2n + 2r\_n(r\_n - 1) + 2s\_n(s\_n - 1)$$

$$\dim\_{\mathbb{F}} Z(\mathfrak{G}^n) = \begin{cases} 6, & \text{if } \quad n = 1 \\ 1 - n + r\_n + s\_{n'} & \text{if } \quad n \gg 2. \end{cases} \tag{16}$$

*where rn and sn are the nth term of the OEIS sequences* A016208 *and* A029858*, respectively.*

**Proof.** Given *n* ∈ N, let P*<sup>n</sup>* = {*A* : *A* ⊆ {1, 2, ... , *n*}}. For *x*, *y* ∈ P*n*, define *x* < *y* if *x* ⊆ *y*. In this case the poset (P*n*, ⊆) consists of all subsets of {1, 2, ... , *n*} ordered by inclusion.

We associate to each finite poset P*<sup>n</sup>* of size *n* the following 2*<sup>n</sup>* × 2*n*-matrix:

$$[M\_{\mathcal{P}\_n}]\_{ij} = \begin{cases} \begin{array}{cc} 1, & if \\ 0, & if \end{array} \begin{array}{c} i, j \text{ are comparable} \\\ \end{array} \end{array} \text{ are comparable}$$

Under appropriate labeling of poset points P*n*, the matrix *M*P*<sup>n</sup>* can be viewed using the Kronecker product as follows:

$$\begin{aligned} M\_{\mathcal{P}\_1} &= \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix} \\ M\_{\mathcal{P}\_2} &= \begin{bmatrix} M\_{\mathcal{P}\_1} & 0 \\ \overline{M\_{\mathcal{P}\_1} & M\_{\mathcal{P}\_1}} \end{bmatrix} = M\_{\mathcal{P}\_1} \otimes M\_{\mathcal{P}\_1} \\ M\_{\mathcal{P}\_3} &= \begin{bmatrix} M\_{\mathcal{P}\_2} & 0 \\ \overline{M\_{\mathcal{P}\_2} & M\_{\mathcal{P}\_2}} \end{bmatrix} = M\_{\mathcal{P}\_1} \otimes M\_{\mathcal{P}\_2} \\ \vdots \\ M\_{\mathcal{P}\_n} &= \begin{bmatrix} M\_{\mathcal{P}\_{n-1}} & 0 \\ \overline{M\_{\mathcal{P}\_{n-1}} & M\_{\mathcal{P}\_{n-1}}} \end{bmatrix} = M\_{\mathcal{P}\_1} \otimes M\_{\mathcal{P}\_{n-1}} \end{aligned}$$

matrices *M*P*<sup>n</sup>* can be seen as pavements, cells with 1's are colored black and those with 0's are colored white. Figure 2 shows examples of these types of matrices.

**Figure 2.** Matrices *M*P*<sup>n</sup>* for *n* = 1, 2, 3 and 4.

*M*P*<sup>n</sup>* is the matrix associated with the polygon *Un* ∈ Γ*<sup>n</sup>* <sup>1</sup> , thus occ(0, *Un*) can be computed in the following fashion:

$$\begin{aligned} \text{occ}(0, lI\_1) &= 1\\ \text{occ}(0, lI\_{\text{ll}}) &= 3(\text{occ}(0, lI\_{n-1})) + 2^{2n-2} \end{aligned} \tag{17}$$

Therefore occ(0, *Un*) = ∑*<sup>n</sup> <sup>k</sup>*=<sup>1</sup> <sup>3</sup>*n*−*k*22(*k*−1) and occ(1, *Un*) = <sup>3</sup>*<sup>n</sup>* thus the result holds.

Now we are interested in computing the trace norm of the {0,1}-Brauer configuration Γ*n*. For this, we recall the following theorem about the singular values of the Kronecker product:

**Theorem 5** ( [36], Theorem 4.2.15)**.** *Let A* ∈ *Mm*,*<sup>n</sup> and B* ∈ *Mp*,*<sup>q</sup> have singular value decompositions A* = *V*1Σ1*W*<sup>∗</sup> <sup>1</sup> *and <sup>B</sup>* <sup>=</sup> *<sup>V</sup>*2Σ2*W*<sup>∗</sup> <sup>2</sup> *and let rankA* = *r*<sup>1</sup> *and rankB* = *r*2*. Then A* ⊗ *B* = (*V*<sup>1</sup> ⊗ *V*2)(Σ<sup>1</sup> ⊗ Σ2)(*W*<sup>1</sup> ⊗ *W*2)∗*. The nonzero singular values of A* ⊗ *B are the r*1*r*<sup>2</sup> *positive numbers* {*σi*(*A*)*σj*(*B*) : 1 *i r*1, 1 *j r*2} *(including multiplicities).*

The following Lemma 1 is helpful to prove Theorem 6.

**Lemma 1.** *Let A* ∈ *Mn*(C) *be a given matrix. If B* = *A* 0 *A A* ∈ *M*2*n*(C) *then the singular values of <sup>B</sup> are φσi*(*A*) *and <sup>φ</sup>*−1*σi*(*A*) *for <sup>i</sup>* <sup>=</sup> 1, ... , *n, where <sup>φ</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> <sup>√</sup><sup>5</sup> <sup>2</sup> *is the golden ratio.*

**Proof.** Note that *B* = *A* 0 *A A* = 1 0 1 1 ⊗ *A*. The singular values for

 1 0 1 1 are *φ* and *φ*<sup>−</sup>1, then by Theorem 5 the result holds.

**Theorem 6.** *For each n* - 1*, if MR*(Γ*n*) = *M*P*<sup>n</sup> is the matrix associated with the polygon Un* ∈ Γ*<sup>n</sup>* <sup>1</sup> *then*

$$\|M\_{\mathcal{P}\_\mathbf{n}}\|\_\* = \mathbb{S}^{\mathbf{n}/2} \tag{18}$$

**Proof.** By induction on *<sup>n</sup>*. For *<sup>n</sup>* <sup>=</sup> 1, *<sup>M</sup>*P<sup>1</sup> <sup>∗</sup> <sup>=</sup> *<sup>φ</sup>* <sup>+</sup> *<sup>φ</sup>*−<sup>1</sup> <sup>=</sup> <sup>√</sup>5. Let us suppose that *M*P*<sup>n</sup>* ∗ = (2*φ* − 1)*<sup>n</sup>* = 5*n*/2 and let us see that the result is fulfilled for *n* + 1, i.e.,

$$\|\|M\_{\mathcal{P}\_n}\|\|\_{\*} = (2\phi - 1)^{n+1} = 5^{\frac{n+1}{2}}$$

Since *M*P*n*+<sup>1</sup> = *M*P<sup>1</sup> ⊗ *M*P*<sup>n</sup>* , then for the Lemma 1 the singular values of *M*P*n*+<sup>1</sup> are

$$\phi \sigma\_i(M\_{\mathcal{P}\_n}) \text{ and } \phi^{-1} \sigma\_i(M\_{\mathcal{P}\_n})$$

for *i* = 1, . . . , 2*n*. Thus,

$$\begin{aligned} \|M\_{\mathcal{P}\_{n+1}}\|\_{\*} &= \sum\_{i=1}^{2^{n+1}} \sigma\_i(M\_{\mathcal{P}\_{n+1}}) \\ &= \sum\_{i=1}^{2^n} \Phi \sigma\_i(M\_{\mathcal{P}\_n}) + \sum\_{i=1}^{2^n} \Phi^{-1} \sigma\_i(M\_{\mathcal{P}\_n}) \\ &= \phi \|M\_{\mathcal{P}\_n}\|\_{\*} + \phi^{-1} \|M\_{\mathcal{P}\_n}\|\_{\*} \\ &= \|M\_{\mathcal{P}\_n}\|\_{\*} \left(\phi + \phi^{-1}\right) = \|M\_{\mathcal{P}\_n}\|\_{\*} (2\phi - 1) \\ &= (2\phi - 1)^{n+1} = 5^{\frac{n+1}{2}} \end{aligned}$$

**Corollary 5.** <sup>∞</sup>

$$\sum\_{n=2}^{\infty} \frac{1}{\|M\_{\mathcal{P}\_n}\|\_{\*}} = \frac{1}{2(3-\phi)}$$

**Proof.** By Theorem 6, we have:

$$\sum\_{n=2}^{\infty} \frac{1}{||M\_{\mathcal{P}\_n}||\_\*} = \sum\_{n=2}^{\infty} \frac{1}{(2\phi - 1)^n}$$

which is a convergent geometric series with *r* = <sup>1</sup> (2*φ*−1) <sup>&</sup>lt; 1 and *<sup>a</sup>* <sup>=</sup> <sup>1</sup> (2*φ*−1)<sup>2</sup> , therefore:

$$\sum\_{n=2}^{\infty} \frac{1}{||M\_{\mathcal{P}\_n}||\_\*} = \frac{\frac{1}{(2\phi - 1)^2}}{1 - \frac{1}{2\phi - 1}} = \frac{1}{2(3 - \phi)^2}$$

3. For *n* - 1 fixed, let Φ*<sup>n</sup>* = {Φ*<sup>n</sup>* <sup>0</sup> , <sup>Φ</sup>*<sup>n</sup>* <sup>1</sup> , *μ*, O} be a {0,1}-Brauer configuration such that:

$$\begin{aligned} \Phi\_0^n &= \{0, 1\}. \\ \Phi\_1^n &= \{\mathcal{U}\_1, \mathcal{U}\_2, \dots, \mathcal{U}\_n\}, \quad \text{for} \ 1 \le i \le n, \quad |\mathcal{U}\_i| = (i+5)^2. \\ \mu(0) &= \mu(1) = 1. \end{aligned} \tag{19}$$

For *i* ≥ 1, the word *wi* associated with the polygon *Ui* has the form *wi* = *wi*,1*wi*,2 ... *wi*,*δ<sup>i</sup>* , *wi*,*<sup>j</sup>* ∈ {0, 1}, *occ*(0, *Ui*)=(*i* + 5)(*i* + 3), *occ*(1, *Ui*) = 2(*i* + 5).

The orientation O is defined in such a way that for successor sequences associated with vertices 0 and 1, it holds that *Ui* < *Ui*+1.

Polygons *Ui* can be seen as (*i* +5) × (*i* +5)-matrices over Z2. Each row *Rj* is defined by coefficients of a polynomial *P<sup>i</sup> <sup>j</sup>*(*t*) with the form *<sup>P</sup><sup>i</sup> <sup>j</sup>*(*t*) = *<sup>u</sup><sup>i</sup> <sup>j</sup>*,1 + *<sup>u</sup><sup>i</sup> <sup>j</sup>*,2*<sup>t</sup>* + ··· + *<sup>u</sup><sup>i</sup> <sup>j</sup>*,*i*+4*t <sup>i</sup>*<sup>+</sup>4, *ui <sup>j</sup>*,*<sup>k</sup>* ∈ {0, 1}.

$$\mathcal{U}\_{1} = \begin{bmatrix} 0 & 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix}$$

$$u^{i}\_{j,k} = u^{i-1}\_{j,k} \quad 1 \le j, k \le i+4,\tag{20}$$

$$u^{i}\_{j+5} = 0, \quad 1 \le j \le i+3,$$

$$u^{i}\_{i+4,i+5} = 1,$$

$$u^{i}\_{i+5,i+4} = 1,$$

$$u^{i}\_{i+5,i+5} = 0.$$

**Theorem 7.** *For n* - 1*, if* F*<sup>n</sup>* = F*Q<sup>n</sup>* Φ*n*/*I<sup>n</sup>* <sup>Φ</sup>*<sup>n</sup> is the Brauer configuration algebra induced by the {0,1}-Brauer configuration* <sup>Φ</sup>*n, <sup>α</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>(*tn*+<sup>5</sup> <sup>−</sup> <sup>6</sup>)*, and <sup>β</sup><sup>n</sup>* <sup>=</sup> *<sup>ε</sup>n*+<sup>5</sup> <sup>−</sup> *<sup>ε</sup>*5*, with <sup>ε</sup><sup>i</sup>* <sup>=</sup> *<sup>i</sup>*(*i*+1)(2*i*+6) <sup>6</sup> *for i* ≥ 1 *then the following statements hold:*

*1.* dim<sup>F</sup> F*<sup>n</sup>* = 2*n* + 2*tαn*−<sup>1</sup> + 2*tβn*−1*,*

$$2. \quad \dim\_{\mathbb{F}} Z(\mathfrak{F}^n) = 1 + n + \varepsilon\_{n+4} - 2n,$$

*3.* Lim*n*→<sup>∞</sup> *<sup>ρ</sup>*(*MR*(Φ*n*)) = 2 + 2 √2*.*

**Proof.** The Formulas (1) and (2). for the dimension of the algebra F*<sup>n</sup>* and its center *Z*(F*n*) are consequences of the definition of a Brauer configuration Φ*<sup>n</sup>* and Corollary 1.

Let us prove identity 3. Firstly, we note that the characteristic polynomials *Pn*(*λ*) associated with matrices *Un* can be obtained recursively. They obey the following general rules according to the size of the corresponding matrices.

$$\begin{aligned} P\_3(\lambda) &= \lambda^3 - 2\lambda, \\ P\_4(\lambda) &= \lambda^4 - 4\lambda^2, \\ P\_n(\lambda) &= \sum\_{j=1}^n a\_j^n \lambda^j, \quad \text{if } n \ge 5, \\ a\_n^n &= 1, \quad a\_{n-1}^n = 0, \quad a\_1^n = (-1)^{n+1} 2, \\ a\_s^n &= a\_{s-1}^{n-1} - a\_s^{n-2}, \quad \text{for the remaining vertices.} \end{aligned}$$

*P*3(*λ*), *P*4(*λ*) and *P*5(*λ*) are characteristic polynomials of the following matrices *T*3, *T*4, and *T*5, respectively:

$$T\_3 = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}, \quad T\_4 = \begin{bmatrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}, \quad T\_5 = \begin{bmatrix} 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{bmatrix}.$$
  $\text{For } s > 0, P(1) \text{ is the characteristic normal manifold of } I(I) \le c \text{ iff }$ 

For any *k* ≥ 6, *Pk*(*λ*) is the characteristic polynomial of *Uk*<sup>−</sup><sup>5</sup> ∈ Φ*n*. We note that for *k* ≥ 5, | 2 + 2 <sup>√</sup><sup>2</sup> <sup>−</sup> *<sup>ρ</sup>*(*MR*(Φ(2*k*−1)))| ≤ <sup>1</sup> <sup>10</sup>*δ<sup>k</sup>* , where

$$
\delta\_k = \begin{cases}
\lceil s\_k \sqrt{2} L n (2^k - 1) \rceil, & \text{if } k \text{ is odd,} \\
\lfloor s\_k \sqrt{2} L n (2^k - 1) \rfloor, & \text{if } k \text{ is even.}
\end{cases}
$$

$$s\_k = \begin{cases} k - 4, & \text{if } \ 5 \le k \le 7, \\ 62^{k-8}, & \text{if } \ k \ge 8. \end{cases}$$

Then Lim *k*→∞ | 2 + 2 <sup>√</sup><sup>2</sup> <sup>−</sup> *<sup>ρ</sup>*(*MR*(Φ(2*k*−1)))<sup>|</sup> <sup>=</sup> 0. Thus, *<sup>ρ</sup>*(*MR*(Φ(2*k*−1))) is a Cauchy subsequence of the sequence *ρ*(*MR*(Φ*n*), *n* ≥ 5 converging to 2 + 2 √2.

**Corollary 6.** *For any n* ≥ 5*, an n-vertex quiver Qn with underlying graph Qn of the form: . . .*

*is of wild type.*

**Proof.** Since *ρ*(*Q*5) = √√17+<sup>5</sup> <sup>2</sup> , then the result holds as a consequence of Theorem 2, Remark 1, and Theorem 7.

The following results [37] regarding some relationship between graph operations and energy graph allow finding upper and lower bounds for *MR*(Φ*n*) ∗.

**Theorem 8** (Theorema 4.18 [37])**.** *Let G, H, and G* ◦ *H be graphs as specified above. Then*

$$\|\|G \diamond H\|\|\_{\*} \lesssim \|\|G\|\|\_{\*} + \|\|H\|\|\_{\*}$$

*Equality is attained if and only if either u is an isolated vertex of G or v is an isolated vertex of H or both.*

**Corollary 7** (Corollary 4.6 [37])**.** *If* {*e*} *is a cut edge of a simple graph G, then G* − {*e*} ∗ < *G* ∗*.*

As a consequence of these results, we obtain the following Corollary 8.

**Corollary 8.** *For n* -6*.*

$$2\sqrt{n-1} < \|M\_R(\Phi^{n-5})\|\_\* < 2 + \begin{cases} 2\csc(\frac{\pi}{2(n-2)}), & \text{if } \quad n-3 \equiv 0(\text{mod } 2), \\\\ 2\cot(\frac{\pi}{2(n-2)}), & \text{if } \quad n-3 \equiv 1(\text{mod } 2). \end{cases} \tag{21}$$

**Proof.** The inequality at right hand holds as a consequence of Theorem 8 taking into account that *Qn* is the coalescence [37] between the cycle C<sup>4</sup> and A*n*−3, and that:

$$\|\mathbb{C}\_4\|\_\* = 4 \text{ and } \|\mathbb{A}\_{n-3}\|\_\* = \begin{cases} 2\csc(\frac{\pi}{2(n-2)}) - 2, & \text{if } \quad n-3 \equiv 0 \text{(mod 2)}, \\\\ 2\cot(\frac{\pi}{2(n-2)}) - 2, & \text{if } \quad n-3 \equiv 1 \text{(mod 2)}. \end{cases}$$

To prove the left hand inequality, we remove edges *c*<sup>1</sup> and *c*<sup>2</sup> in *Qn*, obtaining in this fashion a connected tree. Since among all trees of order *n*, *Sn* attains the minimal energy. The result holds as a consequence of Corollary 7.

#### **4. Concluding Remarks**

{0, 1}-Brauer configuration algebras give rise to the so-called trace norm of a Brauer configuration. Such Brauer configurations are a source of a great variety of graphs and posets via its reduced message. The structure of the adjacency matrices associated with these graphs allows estimating the corresponding trace norm or graph energy values. In line with the main problem in the graph energy theory, we give explicit formulas for the trace norm of some (0, 1)-matrices associated with these families of graphs and posets. On the other hand, bounds for the energy of some families of graphs can be obtained via graph coalescence. It is worth pointing out that some of these graphs underlying quivers of wild type.

An interesting task for the future will be to find the trace norms of a wide variety of Brauer configuration algebras.

**Author Contributions:** Investigation, N.A.M., A.M.C., P.F.F.E. and I.D.M.G.; writing—review and editing, N.A.M., A.M.C., P.F.F.E. and I.D.M.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** MinCiencias-Colombia and Seminar Alexander Zavadskij on Representation of Algebras and their Applications, Universidad Nacional de Colombia.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** N. Agudelo and A.M. Cañadas thanks to MinCiencias and Universidad Nacional de Colombia, sede Bogotá (Convocatoria 848- Programa de estancias Postdoctorales 2019) for their support.

**Conflicts of Interest:** The authors declare no conflict of interest.
