**1. Introduction**

The impact of the study of Laplacian spectra for graphs has increased due to its applications in different fields. "Laplacian spectra have miscellaneous applications in graph theory, combinatorial optimization, mathematical biology, computer science, machine learning and in differential geometry, as well. Due to the wide range of applications, the Laplacian spectra of networks is a very interesting and attractive field of research. Computations of the Laplacian spectra of networks are involved in many results related to topological structures and dynamical processes."

" Arenas et al. [1] developed a method for understanding synchronization phenomena in networks using Laplacian spectra. Synchronization processes in populations of locally-interacting elements are the focus of intense research in physical, biological, chemical, technological and complex network topologies.

Boccaletti et al. [2] focused on coupled biological and chemical systems, neural networks, socially-interacting species, the Internet and the World Wide Web, which are only a few examples of systems composed of a large number of highly interconnected dynamical units. The first approach to capture the global properties of such systems is to model them as graphs whose nodes represent the dynamical units and whose links stand for the interactions between them.

Liu et al. [3] discussed and investigated the properties of the Laplacian matrices for n-prism networks. They calculated the Laplacian spectra of n-prism graphs, which are both planar and polyhedral. In particular, they derived the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Moreover, these results were used to handle various problems that often arise in the study of networks including the Kirchhoff index, global mean-first passage time, average path length and the number of spanning trees.

Ding et al. [4] discussed the Laplacian spectra of a three-prism graph and applied them. This graph is both planar and polyhedral and belongs to the generalized Petersen graph. Using the regular structures of this graph, they obtained the recurrent relationships for the Laplacian matrix between this graph and its initial state of a triangle and further derived the corresponding relationships for Laplacian eigenvalues between them. By these relationships, they obtained the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Finally, they applied these expressions to calculate the number of spanning trees and mean first-passage time (MFPT) and saw the scaling of MFPT with the network size *n*, which is larger than those performed on some uniformly recursive trees.

Therefore, it is of grea<sup>t</sup> interest to compute the Laplacian spectra of different networks. In the last decades, networks and applications of Laplacian spectra have been studied by many scientists, i.e., [5]."

"Researchers have not paid much attention to applications of Laplacian spectra for networks based on different types of graph operations. Since graph products have a very significant contribution to describing very useful complex networks, we have considered networks based on a categorical product. In this paper, we study the Laplacian spectra of the complex network as a categorical product network. Categorical graph products are used to study complex networks in computer science, to understand structures in structural mechanics and to describe multilayer networks and have many applications in network topologies. Considering the structure of categorical product networks, we derive the expressions for the product and sum of reciprocals of nonzero eigenvalues of the categorical product with the help of the eigenvalues of the path and cycle. Furthermore, we compute the Kirchhoff index, global mean first passage time, average path length and number of spanning trees using the relation of nonzero eigenvalues to these applications."

"The Kirchhoff index *K f* , also simply called the resistance and denoted by R [6], of a connected graph *G* on *n* nodes is defined by:

$$\mathcal{K}f = \frac{1}{2} \sum\_{i=1}^{n} \sum\_{j=1}^{n} (\Omega)\_{ij} \tag{1}$$

where (Ω)*ij* is the resistance distance matrix. This formula for the Kirchhoff index reduces to [7]:

$$Kf(G) = N \sum\_{i=2}^{N} \frac{1}{\nu\_i} \tag{2}$$

where *νi* represents the eigenvalue of the Laplacian matrix of the graph.

The global mean-first passage time (MFPT) *Fij* has been studied with respect to transport and networks. The mean first passage time (MFPT) is very useful to estimate the speed of transport for random walks on complex networks [8,9]. In fact, MFPT denoted by < *FN* > measures the diffusion efficiency of random walks, which is obtained by averaging *Fij* over (*N* − 1) possible destinations and N origins of particles:"

$$ = \frac{1}{N(N-1)} \sum\_{i \neq j} F\_{ij} \text{(N)}\tag{3}$$

" From ([10]), let commuting time *Cij* between nodes *i* and *j* be exactly 2 | *E* | *rij*, then we have:

$$\mathcal{L}\_{\vec{i}\vec{j}} = F\_{\vec{i}\vec{j}} + F\_{\vec{j}\vec{i}} = \mathcal{2} \mid E \mid r\_{\vec{i}\vec{j}} \tag{4}$$

" where | *E* | denotes the number of edges in G and *rij* is the effective resistance between two nodes *i* and *j*. By combining the two above relations, the MFPT can be computed by the following formula:"

$$1 < F\_N > = \frac{2E\_t}{N\_t(N\_t - 1)} \sum\_{i < j}^n r\_{ij}(G) = \frac{2E\_t}{N\_t - 1} \sum\_{k=2}^{N\_t} \frac{1}{\nu\_k} \tag{5}$$

where *νk* are the eigenvalues of the Laplacian matrix of the graph after t iterations.

The average path length defines the average number of steps along the shortest path *dij* for all possible pairs of network nodes, which is the measure of the efficiency of information or mass transport on the network, then the average path length *Dt*, for G(t) (the graph after t iterations) is defined as:"

$$D\_t = \frac{2}{N\_t - 1} \sum\_{j \ge i} d\_{ij} \tag{6}$$

" Moreover, the shortest path *dij* and effective resistance *rij* are related by expression *rij* = 2*dij* |*N*| , where N represents the number of nodes in the complete graph. Then, by these two relations, we have:

$$D\_{l} = \frac{2}{N\_{l}(N\_{l} - 1)} \frac{N\_{l}}{2} \sum\_{j>i} r\_{ij} = \frac{N\_{l}}{N\_{l} - 1} \frac{N\_{l}}{2} N\_{l} \sum\_{k=2}^{N\_{l}} \frac{1}{\nu\_{k}} \tag{7}$$

Spanning trees are very important in complex networks and play a key role in various networks. The exact number of spanning trees *Nst*(*Gt*), for *<sup>G</sup>*(*t*), (*t* ≥ <sup>1</sup>), where *t* shows the iterations in constructing a graph, discussed in the next section, can be computed by Kirchhoff's matrix tree theorem [11]. "

" More precisely, in this paper, we have computed Laplacian spectra for categorical product networks and have discussed their applications. After the Introduction, in the second section, the materials and methods are discussed; in the main section, the categorical product network is defined in an iterative way, and then, the Laplacian spectra are computed. Finally, the applications of the Laplacian spectra are computed."
