Products and tensor products of graphs and homomorphisms

We introduce and study, for a process P delivering edges on the Cartesian product of the vertex sets of a given set of graphs, the P-product of these graphs, thereby generalizing many types of product graph. Analogous to the notion of a multilinear map (from linear algebra), a P-morphism is introduced and utilised to define a P-tensor product of graphs, after which its uniqueness is demonstrated. Congruences of graphs are utilised to show a way to handle projections (being weak homomorphisms) in this context. Finally, the graph of a homomorphism and a P-tensor product of homomorphisms are introduced, studied, and linked to the P-tensor product of graphs.