Incidence Hypergraphs: Box Products & the Laplacian

TL;DR

This paper explores the properties of the box product and exponential in various hypergraph categories, introducing a new incidence hypergraph product that unifies graph and hypergraph Laplacian concepts.

Contribution

It introduces a novel incidence hypergraph box product that unifies graph and hypergraph Laplacian frameworks through a complex number analogy.

Findings

Characterization of box exponential for quivers within their category

Introduction of an incidence dual-closed hypergraphic box product

Connection between box exponential evaluated at paths and hypergraphic Laplacian entries

Abstract

The box product and its associated box exponential are characterized for the categories of quivers (directed graphs), multigraphs, set system hypergraphs, and incidence hypergraphs. It is shown that only the quiver case of the box exponential can be characterized via homs entirely within their own category. An asymmetry in the incidence hypergraphic box product is rectified via an incidence dual-closed generalization that effectively treats vertices and edges as real and imaginary parts of a complex number, respectively. This new hypergraphic box product is shown to have a natural interpretation as the canonical box product for graphs via the bipartite representation functor, and its associated box exponential is represented as homs entirely in the category of incidence hypergraphs; with incidences determined by incidence-prism mapping. The evaluation of the box exponential at paths is shown to correspond to the entries in half-powers of the oriented hypergraphic signless Laplacian matrix.