Oriented Hypergraphic Matrix-tree Type Theorems and Bidirected Minors via Boolean Order Ideals

TL;DR

This paper extends matrix-tree theorems to oriented hypergraphs and bidirected graphs using Boolean order ideals, revealing new algebraic and combinatorial insights into graph minors and spanning trees.

Contribution

It introduces a new framework connecting hypergraphic minors, Boolean lattices, and incidence maps, generalizing classical matrix-tree theorems to broader graph classes.

Findings

Determinant formula for signed graphic Laplacian derived from maximal positive-circle-free elements.

Spanning trees characterized as single-element order ideals within Boolean lattices.

Establishment of a correspondence between minors and principal order ideals in signed boolean lattices.

Abstract

Restrictions of incidence-preserving path maps produce an oriented hypergraphic All Minors Matrix-tree Theorems for Laplacian and adjacency matrices. The images of these maps produce a locally signed graphic, incidence generalization, of cycle covers and basic figures that correspond to incidence-k-forests. When restricted to bidirected graphs the natural partial ordering of maps results in disjoint signed boolean lattices whose minor calculations correspond to principal order ideals. As an application, (1) the determinant formula of a signed graphic Laplacian is reclaimed and shown to be determined by the maximal positive-circle-free elements, and (2) spanning trees are equivalent to single-element order ideals.