*4.1. Kirchhoff Index*

" The Kirchhoff index of a connected graph *G* is defined as the sum of resistance distances between all pairs of vertices, mathematically:"

$$Kf(G(t)) \ = \sum\_{i$$

where *rij*(*G*) represents the resistance distance between a pair of vertices. In terms of eigenvalues for a connected network *G* of order *N* with all its nonzero eigenvalues represented by *νi*; *i* = 2, ... , *N*, a well-known identity Kirchhoff index is defined as:

$$Kf(G(t)) \ = \ \ N\_t \sum\_{k=2}^{N\_t} \frac{1}{\nu\_k} = N\_t \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\nu\_{i,j}} \quad (i,j) \neq (0,0)$$

The Kirchhoff index for the categorical path-path product network:"

$$\begin{aligned} \mathcal{K}f(G(t)) &= \ & \text{at} \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{\left( (2 - 2\cos\frac{j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{j\pi}{n})(2 - 2\cos\frac{i\pi}{t}) \right)} \\ (i, j) & \neq & (0, 0) \end{aligned}$$

The Kirchhoff index for the categorical cycle-path product network:

$$\mathcal{K}f(G(t)) \quad = \ \text{at} \\ \sum\_{i=0}^{t-1} \sum\_{j=0}^{n-1} \frac{1}{(2 - 2\cos\frac{2j\pi}{n})d\_{i+1} + d\_{j+1}(2 - 2\cos\frac{i\pi}{t}) - (2 - 2\cos\frac{2j\pi}{n})(2 - 2\cos\frac{i\pi}{t})}$$

The Kirchhoff index for the categorical cycle-cycle network:

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