Intersection graphs of oriented hypergraphs and their matrices

For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. Continuing the study of these matrices associated to an oriented hypergraph, several related structures are investigated including: the incidence dual, the intersection graph (line graph), and the 2-section. The intersection graph is shown to be the 2-section of the incidence dual. Also, any simple oriented signed graph is the intersection graph of an infinite family of oriented hypergraphs. Matrix relationships between these new constructions are also established, which lead to new questions regarding associated eigenvalue bounds. Vertex and edge-switchings on these various structures are also studied. A connection is then made between oriented hypergraphs and balanced incomplete block designs.