Grassmann Integral Representation for Spanning Hyperforests

TL;DR

This paper introduces a Grassmann algebra approach to hypergraphs, providing a unified framework for spanning hyperforests and hypertrees, generalizing Kirchhoff's matrix-tree theorem and revealing supersymmetry structures.

Contribution

It develops a Grassmann integral representation for spanning hyperforests, extending classical graph theory results to hypergraphs with supersymmetry insights.

Findings

Derived generating functions for hyperforests and hypertrees

Generalized Kirchhoff's matrix-tree theorem to hypergraphs

Identified supersymmetry in the integral representation

Abstract

Given a hypergraph G, we introduce a Grassmann algebra over the vertex set, and show that a class of Grassmann integrals permits an expansion in terms of spanning hyperforests. Special cases provide the generating functions for rooted and unrooted spanning (hyper)forests and spanning (hyper)trees. All these results are generalizations of Kirchhoff's matrix-tree theorem. Furthermore, we show that the class of integrals describing unrooted spanning (hyper)forests is induced by a theory with an underlying OSP(1|2) supersymmetry.