extension being made in this paper is to generalize graph product operation to arbitrary linear combination of Cartesian and direct products. Specifically, we define the following [ *generalized product*]{}:

Generalized product $\displaystyle G \operatorname*{\boxtimes}_{\alpha, \beta} H$

:   A network that is obtained as a weighted linear combination of $G \operatorname*{\Box}H$ and $G \operatorname*{\times}H$, where $\alpha$ and $\beta$ are used as weights. Its adjacency matrix is given by $$A_{G \operatorname*{\boxtimes}_{\alpha, \beta} H} = A_G \operatorname*{\mathrlap{\oplus}\otimes}_{\alpha, \beta} A_H = \alpha A_G \operatorname*{\oplus}A_H + \beta A_G \operatorname*{\otimes}A_H, \label{eq:gp}$$

where the symbol $\displaystyle\operatorname*{\mathrlap{\oplus}\otimes}_{\alpha, \beta}$ is newly introduced in this paper to represent a weighted sum of the Kronecker sum and product. This graph product operation is still commutative and associative. All of CP/DP/SPMNs can be described uniformly using this generalized product ($\displaystyle\operatorname*{\boxtimes}_{1,0} = \operatorname*{\Box}$, $\displaystyle\operatorname*{\boxtimes}_{0,1} = \operatorname*{\times}$, and $\displaystyle\operatorname*{\boxtimes}_{1,1} =
\operatorname*{\boxtimes}$). Moreover, setting $\alpha$ and/or $\beta$ to non-integer values represents a more nuanced balance between Cartesian and direct products in the GPMN structure (note that this is made possible by the first extension). We call networks generated by using this generalized product [*generalized GPMNs (GGPMNs)*]{}.

We have found that the strength (degree), adjacency, and Laplacian spectra of GGPMNs can be given exactly or approximately 