**Shin Min Kang 1,2,***∗***, Muhammad Kamran Siddiqui 3,4, Najma Abdul Rehman 3, Muhammad Imran 4,5 and Mehwish Hussain Muhammad 6**


Received: 26 May 2018; Accepted: 6 June 2018; Published: 7 June 2018

**Abstract:** The Kirchhoff index, global mean-first passage time, average path length and number of spanning trees are of grea<sup>t</sup> importance in the field of networking. The "Kirchhoff index" is known as a structure descriptor index. The "global mean-first passage time" is known as a measure for nodes that are quickly reachable from the whole network. The "average path length" is a measure of the efficiency of information or mass transport on a network, and the "number of spanning trees" is used to minimize the cost of power networks, wiring connections, etc. In this paper, we have selected a complex network based on a categorical product and have used the spectrum approach to find the Kirchhoff index, global mean-first passage time, average path length and number of spanning trees. We find the expressions for the product and sum of reciprocals of all nonzero eigenvalues of a categorical product network with the help of the eigenvalues of the path and cycles.

**Keywords:** Laplacian spectra; categorical product; Kirchhoff index; global mean-first passage time; spanning tree

**MSC:** 05C12, 05C90
