**6. Conclusions**

In this paper, we discuss a complex network based on the categorical product and have used the spectrum approach to find the Kirchhoff index. The global mean first-passage time (MFPT) is computed for a complex network based on the categorical product. Moreover, using the Laplacian spectra for the categorical product network, we compute the average path length, which is the basic idea in network topologies. It describes the measure of the efficiency of transport (mass or information) on a network. The last application of Laplacian spectra for the categorical product network that we have computed was for spanning trees, which is the direct application in designing a network. We can extend our work to different networks based on graph operations other than the categorical product."

These results are indirectly related to entropy. Using the same spectra, we have future plans to compute the global first-passage time for maximal-entropy random walks in categorical product networks, and the categorical product network entropy is also in our future plans using the idea given in [19,20].

**Author Contributions:** S.M.K. contribute for conceptualization, funding, and analyzed the data. M.K.S. contribute for supervision, methodology, software, validation, designing the experiments and formal analysing. N.A.R. and M.H.M. contribute for performed experiments, resources, some computations and wrote the initial draft of the paper which were investigated and approved by M.K.S. and M.I., and wrote the final draft. All authors read and approved the final version of the paper.

**Acknowledgments:** The authors are grateful to the anonymous referees for their valuable comments and suggestions that improved this paper. This research is supported by the Start-Up Research Grant 2016 of the United Arab Emirates University (UAEU), Al Ain, United Arab Emirates via Grant No. G00002233 and UPAR Grant of UAEU via Grant No. G00002590. Also This research is supported by The Higher Education Commission of Pakistan Under Research and Development Division, National Research Program for Universities via Grant No.: 5282/Federal/NRPU/R&D/HEC/2016.

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