Products and tensor products of graphs and homomorphisms

TL;DR

This paper generalizes various graph products by introducing the P-product, a tensor product of graphs, and explores associated homomorphisms, congruences, and their properties, extending the algebraic framework of graph theory.

Contribution

It introduces the P-product and P-tensor product of graphs, establishing their properties and uniqueness, and connects these concepts with homomorphisms and congruences.

Findings

Defined the P-product as a generalization of many graph products.

Established the existence and uniqueness of the P-tensor product of graphs.

Linked P-tensor products with graph homomorphisms and congruences.

Abstract

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.