*3.1. Categorical Product of Two Paths and Laplacian Spectra*

"Let a categorical product network of paths be constructed in an iterative way. We take categorical product network *<sup>G</sup>*(*t*), (*t* ≥ 1) after *t* − 1 iterations. Initially at *t* = 1, *G*(1) is a path with n vertices. For *t* ≥ 2, *G*(*t*) is constructed from *G*(*t* − <sup>1</sup>); from every existing vertex in *G*(*t* − <sup>1</sup>), a new vertex is created so that a new path with *n* vertices is constructed; also, each new vertex in *G*(*t*) is connected to the vertices in *G*(*t* − <sup>1</sup>), shown by Figure 2. The number of vertices and edges in *G*(*t*) is *Nt* = *nt* and *Et* = (2*t* − <sup>2</sup>)*n* − (2*t* − <sup>2</sup>), *n* ≥ 2."

"Let *λt* and *μt* be the product of all nonzero eigenvalues of *Gt* and the sum of reciprocals of these eigenvalues, respectively, i.e., *λt* = ∏*Nt i*=2 *νi* and *μt* = ∑*Nt i*=2 1*νi* , where *ν*1 = 0 and *νi*, *i* = 2, 3, ... , *Nt* denote the *Nt* − 1 nonzero eigenvalues of *<sup>L</sup>*(*Gt*)."

**Theorem 1.** *The product and sum of reciprocal nonzero eigenvalues of <sup>L</sup>*(*Gt*), *the Laplacian matrix of G(t), are:*

$$\begin{aligned} \lambda\_{l} &= \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{j\pi}{n}) d\_{i+1} + d\_{j+1} (2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right) \\ \mu\_{l} &= \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{j\pi}{n}) d\_{i+1} + d\_{j+1} (2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right)} \end{aligned}$$

*di shows the degree of vertex i.*

**Proof.** " Consider a categorical product network *G* as is shown in the figure. By the properties of the Kronecker product of matrices, we can write the Laplacian matrix for *G* as [15],"

$$L(G\_t) = L(P\_n) \otimes D(P\_t) + D(P\_n) \otimes L(P\_t) - L(P\_n) \otimes L(P\_t)$$

where *D*(*Pn*) is the diagonal matrix of order *n* × *n*; with the diagonal elements' degree of vertices. By using the results from linear algebra, there exists invertible matrices P and Q such that:

$$\left(L(P\_n)\right)' = P^{-1}L(P\_n)P\_\prime \quad \left(L(P\_t)\right)' = Q^{-1}L(P\_t)Q^{-1}$$

are the upper triangular matrices with diagonal elements, 2 − 2 cos *πj n* , *j* = 0, 1, ... , *n* − 1 and 2 − 2 cos *πit*, *i* = 0, 1, . . . , *t* − 1, respectively. Then, using the fact that:

$$(P \otimes Q)^{-1} \cdot (L(P\_n) \otimes L(P\_l)) \cdot (P \otimes Q) = (P^{-1} \otimes Q^{-1}) \cdot (L(P\_n)P \otimes L(P\_l)Q) = P^{-1}L(P\_n)P \otimes Q^{-1}L(P\_l)Q$$

is the upper triangular matrix with diagonal elements, the matrix:"

$$\frac{1}{2}\left(P\otimes Q\right)^{-1}\cdot\left(L(P\_{\mathfrak{n}})\otimes D(P\_{\mathfrak{l}})+D(P\_{\mathfrak{n}})\otimes L(P\_{\mathfrak{l}})-L(P\_{\mathfrak{n}})\otimes L(P\_{\mathfrak{l}})\right)\cdot\left(P\otimes Q\right)^{-1}$$

is upper triangular matrix with diagonal elements,

$$(2 - 2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{t});\tag{8}$$

$$i = 0, 1, \dots, t - 1, \quad j = 0, 1, \dots, n - 1,$$

$$d\_1 = d\_t = d\_n = d\_{nt} = 1,$$

$$d\_{k+1} = d\_{nt-k} = 2, \quad k = 1, \dots, n - 2, \quad d\_{1+nq} = d\_{n+nq} = 2, \quad q = 1, \dots, t - 2,$$

$$\text{all other } d\_l \text{ have a value of four.}$$

that are the eigenvalues for the categorical product network. Therefore:

$$\begin{aligned} \lambda\_t &= \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} v\_{ij} \\ &= \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right) \\ \mu\_t &= \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\nu\_{i,j}} \\ &= \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{j\pi}{t}) \right)} \end{aligned} \tag{10}$$

**Corollary 1.** *" Let <sup>L</sup>*(*<sup>G</sup>*2) *be the Laplacian matrix of Pn* × *P*2*, the categorical product graph with n vertices after the first iteration, t* = 2*, then the product and sum of the reciprocal nonzero eigenvalues of <sup>L</sup>*(*<sup>G</sup>*2) *are:"*

**Figure 2.** (**a**) Path *Pn*. (**b**) Grid *Pn* × *P*2 after the first iteration for *t* = 2. (**c**) General grid *Pn* × *Pt* structure.

### *3.2. Categorical Product of the Cycle-Path and Laplacian Spectra*

Let a categorical product network of the cycle and path be constructed in an iterative way. We take the initial categorical product network *<sup>G</sup>*(*t*), (*t* ≥ 1) after *t* − 1 iterations. Initially, at *t* = 1, *G*(1) is a cycle with *n* vertices. For *t* ≥ 2, *G*(*t*) is constructed from *G*(*t* − <sup>1</sup>), from every existing vertex in *G*(*t* − <sup>1</sup>), a new vertex is created so that a new path with *n* vertices is constructed; also, each new vertex in *G*(*t*) is connected to vertices in *G*(*t* − <sup>1</sup>), shown by Figure 3. The number of vertices and edges in *G*(*t*) are *Nt* = *nt* and *Et* = <sup>2</sup>*n*(*<sup>t</sup>* − <sup>1</sup>), *n* ≥ 2.

Let *λt* and *μt* be the product of all nonzero eigenvalues of *Gt* and the sum of reciprocals of these eigenvalues, respectively, i.e., *λt* = ∏*Nt i*=2 *νi* and *μt* = ∑*Nt i*=2 1*νi* , where *ν*1 = 0 and *νi*, *i* = 2, 3, ... , *Nt* denote the *Nt* − 1 nonzero eigenvalues of *<sup>L</sup>*(*Gt*)."

**Theorem 2.** *The product and sum of reciprocal nonzero eigenvalues of <sup>L</sup>*(*Gt*)*, the Laplacian matrix of G(t), are:*

$$\begin{aligned} \mu\_{t} &= \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right) \\ \mu\_{t} &= \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right)} \end{aligned}$$

**Figure 3.** (**a**) Cycle *Cn*. (**b**) Grid *Cn* × *P*2 after the first iteration for *t* = 2. (**c**) General grid *Cn* × *Pt* structure.

**Proof.** Consider a categorical product network *G* as is shown in the figure. By the properties of the Kronecker product of matrices, we can write the Laplacian matrix for *G* as:"

$$L(\mathsf{G}\_{\mathsf{t}}) = L(\mathsf{C}\_{\mathsf{n}}) \otimes D(\mathsf{P}\_{\mathsf{t}}) + D(\mathsf{C}\_{\mathsf{n}}) \otimes L(\mathsf{P}\_{\mathsf{t}}) - L(\mathsf{C}\_{\mathsf{n}}) \otimes L(\mathsf{P}\_{\mathsf{t}})$$

where *D*(*Cn*) is the diagonal matrix of order *n* × *n*; with the diagonal elements' degree of vertices. By using the results from linear algebra, there exists invertible matrices P and Q such that:

$$\left(L(\mathbb{C}\_{\mathfrak{n}})\right)' = P^{-1}L(\mathbb{C}\_{\mathfrak{n}})P, \quad \left(L(\mathbb{P})\right)' = \mathbb{Q}^{-1}L(\mathbb{P})\mathbb{Q}$$

are the upper triangular matrices with diagonal elements, 2 − 2 cos 2*πj n* , *j* = 0, 1, ... , *n* − 1 and 2 − 2 cos *πit*, *i* = 0, 1, . . . , *t* − 1, respectively. Then, using the fact that:

$$\begin{aligned} \left( (P \otimes Q)^{-1} \cdot (L(\mathbb{C}\_n) \otimes L(P\_l)) \cdot (P \otimes Q) \right) &= (P^{-1} \otimes Q^{-1}) \cdot \left( L(\mathbb{C}\_n) P \otimes L(P\_l) Q \right) \\ &= P^{-1} L(\mathbb{C}\_n) P \otimes Q^{-1} L(P\_l) Q \end{aligned}$$

is the upper triangular matrix with diagonal elements, the matrix:

$$(P \otimes Q)^{-1} \cdot (L(\mathbb{C}\_{\mathfrak{n}}) \otimes D(P\_{\mathfrak{l}}) + D(\mathbb{C}\_{\mathfrak{n}}) \otimes L(P\_{\mathfrak{l}}) - L(\mathbb{C}\_{\mathfrak{n}}) \otimes L(P\_{\mathfrak{l}})) \cdot (P \otimes Q)$$

is the upper triangular matrix with diagonal elements,"

$$(2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{t});\tag{11}$$

$$i = 0, 1, \dots, t - 1, \quad j = 0, 1, \dots, n - 1,$$

$$\text{all } d\_i \text{ have a value of two.}$$

that are the eigenvalues for the categorical product network. Therefore,

$$\begin{array}{rcl} \lambda\_t &=& \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} \nu\_{i,j} \\ &=& \prod\_{i=0}^{t-1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{2j\pi}{n}) d\_{i+1} + d\_{j+1} (2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right) \end{array} \tag{12}$$

$$\begin{array}{rcl} \mu\_{t} &=& \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\nu\_{i,j}} \\ &=& \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{2j\pi}{\pi})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{\pi}) - (2 - 2\cos\frac{2j\pi}{\pi})(2 - 2\cos\frac{i\pi}{\pi}) \right)} \end{array} \tag{13}$$

### *3.3. Categorical Product of the Cycle-Cycle and Laplacian Spectra*

" Let a categorical product network of cycles be constructed in an iterative way. We take initial categorical product network *<sup>G</sup>*(*t*), (*t* ≥ 1) after *t* − 1 iterations. Initially, at *t* = 1, *G*(1) is a categorical product network of cycle *Cn* × *C*3 with n vertices. For *t* ≥ 2, *G*(*t*) is constructed from *G*(*t* − <sup>1</sup>), and from every existing vertex in *G*(*t* − <sup>1</sup>), a new vertex is created so that n vertices are constructed; also, each new vertex in *G*(*t*) is connected to the vertices in *G*(*t* − <sup>1</sup>), shown by Figure 4. The number of vertices and edges in *G*(*t*) are *Nt* = *n*(*<sup>t</sup>* + 2) and *Et* = <sup>2</sup>*n*(*<sup>t</sup>* + 2) − 2."

**Figure 4.** (**a**) Initial product *Cn* × *C*3 for *t* = 1. (**b**) Grid *Cn* × *C*4 after the first iteration for *t* = 2.

Let *λt* and *μt* be the product of all nonzero eigenvalues of *Gt* and the sum of reciprocals of these eigenvalues, respectively, i.e., *λt* = ∏*Nt i*=2 *νi* and *μt* = ∑*Nt i*=2 1*νi* , where *ν*1 = 0 and *νi*, *i* = 2, 3, ... , *Nt* denote the *Nt* − 1 nonzero eigenvalues of *<sup>L</sup>*(*Gt*)."

**Theorem 3.** *The product and sum of the reciprocal nonzero eigenvalues of <sup>L</sup>*(*Gt*)*, the Laplacian matrix of G(t), are:*

$$\begin{array}{rcl} \mu\_{t} &=& \prod\_{i=0}^{t+1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{2i\pi}{t+2}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{2i\pi}{t+2}) \right) \\\\ \mu\_{t} &=& \sum\_{i=0}^{t+1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{2j\pi}{t+2}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{2j\pi}{t+2}) \right)} \end{array}$$

**Proof.** Consider a categorical product network *G* as is shown in the figure. By the properties of the Kronecker product of matrices, we can write the Laplacian matrix for *G* as:"

$$L(\mathsf{G}\_{\mathsf{t}}) = L(\mathsf{C}\_{\mathsf{n}}) \otimes D(\mathsf{C}\_{\mathsf{t}}) + D(\mathsf{C}\_{\mathsf{n}}) \otimes L(\mathsf{C}\_{\mathsf{t}}) - L(\mathsf{C}\_{\mathsf{n}}) \otimes L(\mathsf{C}\_{\mathsf{t}})$$

where *D*(*Cn*) is the diagonal matrix of order *n* × *n*; with the diagonal elements' degree of vertices. By using the results from linear algebra, there exists invertible matrices P and Q such that:

$$\left(L(\mathbb{C}\_{\mathfrak{n}})\right)' = P^{-1}L(\mathbb{C}\_{\mathfrak{n}})P, \quad \left(L(\mathbb{C}\_{\mathfrak{l}})\right)' = \mathbb{Q}^{-1}L(\mathbb{C}\_{\mathfrak{l}})\mathbb{Q}$$

are the upper triangular matrices with diagonal elements, 2 − 2 cos 2*πj n* , *j* = 0, 1, ... , *n* − 1 and 2 − 2 cos 2*πi t*+2, *i* = 0, 1, . . . , *t* + 1, respectively. Then, using the fact that:

$$(P \otimes Q)^{-1} \cdot (L(\mathbb{C}\_n) \otimes L(\mathbb{C}\_t)) \cdot (P \otimes Q) = 0$$

*Symmetry* **2018**, *10*, 206

$$(P^{-1} \otimes Q^{-1}) \cdot (L(\mathbb{C}\_n)P \otimes L(\mathbb{C}\_l)Q) = P^{-1}L(\mathbb{C}\_n)P \otimes Q^{-1}L(\mathbb{C}\_l)Q$$

is the upper triangular matrix with diagonal elements, the matrix:

$$\left( \left( P \otimes Q \right)^{-1} \cdot \left( L\left( \mathbb{C}\_{\mathbb{H}} \right) \otimes D\left( \mathbb{C}\_{\mathbb{H}} \right) + D\left( \mathbb{C}\_{\mathbb{H}} \right) \otimes L\left( \mathbb{C}\_{\mathbb{H}} \right) - L\left( \mathbb{C}\_{\mathbb{H}} \right) \otimes L\left( \mathbb{C}\_{\mathbb{H}} \right) \right) \cdot \left( P \otimes Q \right) \right)$$

is the upper triangular matrix with diagonal elements,"

$$\begin{aligned} &(2-2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{2j\pi}{t+2}) - (2-2\cos\frac{2j\pi}{n})(2-2\cos\frac{2j\pi}{t+2});\\ &i = 0, 1, \dots, t+1, \quad j = 0, 1, \dots, n-1, \\ &d\_1 = d\_t = d\_{tt} = d\_{nt} = 3, \quad \text{all other } d\_l \text{ have a value of four.} \end{aligned} \tag{14}$$

that are the eigenvalues for the categorical product network. Therefore:

$$\begin{array}{rcl} \lambda\_l &=& \prod\_{i=0}^{l+1} \prod\_{j=0}^{n-1} v\_{l,j} \\ &=& \prod\_{i=0}^{l+1} \prod\_{j=0}^{n-1} \left( (2 - 2\cos\frac{2j\pi}{n}) d\_{i+1} + d\_{j+1} (2 - 2\cos\frac{2j\pi}{l+2}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{2j\pi}{l+2}) \right) \end{array} \tag{15}$$

$$\begin{array}{rcl} \mu\_{l} &=& \sum\_{i=0}^{l+1} \sum\_{j=0}^{n-1} \frac{1}{\upsilon\_{l,j}}\\ &=& \sum\_{i=0}^{l+1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2-2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2-2\cos\frac{2j\pi}{1+2}) - (2-2\cos\frac{2j\pi}{n})(2-2\cos\frac{2j\pi}{1+2}) \right)} \end{array} \tag{16}$$
