Incidence Hypergraphs: Injectivity, Uniformity, and Matrix-tree Theorems

TL;DR

This paper develops a comprehensive algebraic and combinatorial framework for oriented hypergraphs, generalizing classical graph concepts to hypergraph incidence matrices, and establishing new matrix-tree and Sachs-coefficient theorems.

Contribution

It introduces incidence hypergraphs with a unifying theory for matrix-tree and Sachs-coefficient theorems, including new characteristic polynomials and subobject classifiers.

Findings

Derived a multivariable all-minors characteristic polynomial for hypergraph matrices.

Established a correspondence between degree-k monomials in the Laplacian and k-arborescences.

Provided a unifying theorem generalizing classical graph theorems to hypergraphs.

Abstract

An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in the subobject classifier of incidence hypergraphs. Moreover, the injective envelope is calculated and shown to contain the class of uniform hypergraphs -- providing a combinatorial framework for the entries of incidence matrices. A multivariable all-minors characteristic polynomial is obtained for both the determinant and permanent of the oriented hypergraphic Laplacian and adjacency matrices arising from any integer incidence matrix. The coefficients of each polynomial are shown to be submonic maps from the same family into the injective envelope limited by the subobject classifier. These results provide a unifying theorem for oriented hypergraphic matrix-tree-type and Sachs-coefficient-type theorems. Finally, by specializing to bidirected graphs, the trivial subclasses for the degree-$k$ monomials of the Laplacian are shown to be in one-to-one correspondence with $k$-arborescences.