On MultiLine Graphs, MultiLine HyperGraphs, and MultiLine Super-HyperGraphs

A finite hypergraph generalizes an ordinary graph by permitting a hyperedge to connect any nonempty subset of vertices, thereby representing genuine multiway interactions. Extending this idea, a finite SuperHyperGraph is obtained through an iterated powerset construction, so that set-valued objects formed at one level may function as vertices or edge endpoints at the next, providing a natural framework for hierarchical and multilayer relational structures. In contrast, a line graph transforms each edge of a graph into a vertex, with two such new vertices adjacent precisely when the corresponding original edges share an endpoint. In this paper, we introduce the notion of a MultiLine Graph, in which multiple edges can be assigned to a vertex, and then develop its higher-order extensions, namely the MultiLine HyperGraph and the MultiLine Super HyperGraph. We further investigate their fundamental properties and structural characteristics.