**2. Materials and Methods**

In this section, we state the materials and methods that are used in the main section.

"In graph theory, the direct product *G* × *H* of graphs G and H is defined as a graph such that the vertex set of *G* × *H* is the Cartesian product *V*(*G*) × *V*(*H*) and any two vertices (u, *u*1) and (v, *v*1) are adjacent in *G* × *H* if and only if u is adjacent with v and *u*1 is adjacent with *v*1. The direct product is also called the tensor product or categorical product.

Let *G* be a graph with vertices 1, 2, ... , *n*. The Laplacian matrix of G is *L*(*G*) = *D*(*G*) − *<sup>A</sup>*(*G*), where A(G) is *n* × *n* adjacency, and the matrix of G with (*i*, *j*) − *entry* is equal to 1 if vertices *i* and *j* are adjacent and 0 otherwise. *D*(*G*) is the diagonal matrix of vertices' degrees."

**Definition 1.** *" Consider two matrices A and B. The Kronecker product A* ⊗ *B of two matrices A and B is the matrix that is obtained by taking* (*i*, *j*)*-th entries as aij<sup>B</sup> for all i*, *j. The Kronecker product of the matrix A* ∈ *M*(*p*,*q*) *with the matrix B* ∈ *M*(*<sup>r</sup>*,*<sup>s</sup>*) *is defined as (see [12,13]):"*

$$
\begin{pmatrix} a\_{11}B & \dots & a\_{1q}B \\ \dots & \ddots & \ddots \\ a\_{p1}B & \dots & a\_{pq}B \end{pmatrix}
$$

The Kronecker product has the following main properties.

$$\begin{array}{rclcrcl}(\mathsf{a}A)\otimes B&=&A\otimes (\mathsf{a}B)=\mathsf{a}(A\otimes B),&\forall\_{\,}A\in M^{(p,q)},B\in M^{(r,s)}\quad\text{and scalar}\;\mathsf{a}A\\(\mathsf{A}\otimes B)^{T}&=&A^{T}\otimes B^{T},&\forall A\in M^{p,q},B\in M^{r,s}\\(\mathsf{A}\otimes B)\otimes \mathbb{C}&=&A\otimes (\mathsf{B}\otimes \mathbb{C}),&\forall A\in M^{(\mathsf{w},\mathsf{n})},B\in M^{(p,q)},\mathsf{C}\in M^{(r,s)}\\(\mathsf{A}\otimes B)(\mathsf{C}\otimes D)&=&A\mathsf{C}\otimes BD,&\forall A\in M^{(p,q)},B\in M^{(r,s)},\mathsf{C}\in M^{(q,\mathsf{k})},D\in M^{(s,l)}\\\text{trace}(\mathsf{A}\otimes B)&=&\text{trace}(\mathsf{B}\otimes A)=\text{trace}(\mathsf{A})\text{trace}(\mathsf{B}),&\forall A\in M^{m},B\in M^{n}.\end{array}$$

"We have used the Kronecker product of matrices to find the Laplacian spectra for the categorical product considering the spectra for the path and cycle. Let *Pn* denote the path with n vertices. Let *Cn* be a cycle of length n. Then, their spectra can be stated as [14]:"

**Lemma 1.** *The Laplacian eigenvalues of a path Pn are* 2 − 2 cos *iπn ,* (*i* = 0, 1, ... , *n* − <sup>1</sup>). *The Laplacian eigenvalues of a cycle Cn are* 2 − 2 cos 2*jπn, j = 0, 1,..., <sup>n</sup>*−*1.*

With the help of the following flowchart in Figure 1, we will facilitate the understanding of the proposed approach in this paper clearly.

**Figure 1.** Flowchart for the methods.
