Functional Product of Graphs: Properties and applications

This paper presents a generalization of the cartesian product of graphs, which we call the functional product of graphs.We provesome properties of this new product, and we show that it is commutative, associative under certain conditions, and it hasa neutral element,which consists of a single vertex without edges (the trivialgraph). We present a characterization of the graphs, which can be obtained fromfunctional product of other graphs. We prove that the maximum degree of the product graph is the sum of the maximum degreesof the factorgraphs, and we present conditions that ensure the connectedness of the product graph. Finally, we present an application of the functionalproduct of graphs, in which we prove some results that allow to generate graphs that admit an equitable total coloring, with at most∆ + 2colors.