have gained a lot of attention in the network science and complex systems science communities over the last several years [@kivela2014multilayer; @boccaletti2014structure; @dedomenico2016]. Multilayer networks describe complex systems in multiple subsystems (layers) and their interconnectivities. This offers an intuitive, powerful, and practical framework for modeling and analysis of complex systems.

In the multilayer networks literature, [*graph product*]{}, a multiplication of two or more graphs discussed in discrete mathematics, is often used as a representation of multilayer network topology [@leskovec2010kronecker; @sole2013spectral; @de2013mathematical; @asllani2015turing; @brechtel2016master]. It is sometimes used explicitly and visually (e.g., Cartesian product of two graphs), or at other times implicitly through application of non-conventional product to matrices (e.g., Kronecker product of adjacency matrices). To the best of our knowledge, however, there is an apparent lack of systematic references that summarize various properties of such graph-product-based multilayer network topologies and their implications for the structure and dynamics of complex networks. The present study aims to meet this need.

In this paper, we aim to establish theoretical foundations of graph product multilayer networks (GPMNs), a family of multilayer networks that can be obtained as a graph product of two or more factor networks. We primarily consider the following three major graph 