School of Mathematical Sciences

We describe four types of hyperspace graphs; namely, the simultaneous and nonsimultaneous symmetric product graphs, as well as their respective layers. These hyperspace graphs are meant to be analogous to the concepts of hyperspaces in topology, in that they are constructed by taking in another graph as an input in the construction of the hyperspace graph. We establish subgraph relationship between these graphs and establish some properties on the orders and sizes of the graphs, as well as on the degrees of the individual vertices of these graphs. We establish that these graphs are connected (providing that the input graph is connected), and provide a categorization of the graphs G for which the second symmetric product graphs are planar. We investigate the chromatic numbers and hamiltonicity of some of these graph products. We also provide a categorization for the distances between any pair of vertices in the symmetric product graphs. We conclude by discussing a couple of different unanswered questions that could be addressed in the future