On the Embeddings of the Semi-Strong Product Graph

Abstract

Over the years, a lot has been written about the three more common graph products (Cartesian product, Direct product and the Strong product), as all three of these are commutative products. This thesis investigates a non-commutative product graph, H, G, we call the Semi-Strong graph product, also referred in the literature as the Augmented Tensor and/or the Strong Tensor. We will start by discussing its basic properties and then focus on embeddings where the second factor, G, is a regular graph. We will use permutation voltage graphs and their graph coverings to compute the minimum genus for several families of graphs. The results follow work started first by A T White [12], extended by Ghidewon Abay Asmerom [1],[2], and follows the lead of Pisanski [9]. The strategy we use starts with an embedding of a graph H and then modifying H creating a pseudograph H*. H* is a voltage graph whose covering is HxG. Given the graph product HxG, where G is a regular graph and H meets certain conditions, we will use the embedding of H to study topological properties, particularly the surface on which HxG is minimally embedded