Tensor SuperHyperGraph and Tensor Iterated Line Graph

A hypergraph extends an ordinary graph by permitting each edge to join an arbitrary nonempty subset of the vertex set. By iterating the powerset construction one additional level, one obtains nested (higher-order) vertex objects and, consequently, finite SuperHyperGraphs whose vertices and edges can themselves be set-valued across multiple layers. A line graph is a transformation that replaces each edge of a graph with a vertex, connecting two such vertices if and only if the corresponding edges in the original graph share a common endpoint. A tensor graph can be viewed as a multiplex network: an r-tuple of graphs on a common vertex set, encoded by stacking their adjacency matrices into a single third-order adjacency tensor. In this paper, we introduce the notion of a Tensor SuperHyperGraph. Using the line-graph transformation, we further define the Tensor Line Graph and the Tensor Iterated Line Graph, and we investigate fundamental properties of these constructions.