Laplacian matrices and spanning trees of tree graphs

TL;DR

This paper investigates the structure of Laplacian matrices and spanning trees of directed graphs, revealing a factorization of determinants of lifted Schr"odinger operators into products related to subgraphs, with combinatorial insights and applications.

Contribution

It introduces a novel factorization of determinants of lifted Schr"odinger operators on the graph of spanning trees, with a combinatorial description of multiplicities and block structures.

Findings

Determinant of lifted Schr"odinger operator factors into subgraph determinants.

Provides a combinatorial exploration procedure for multiplicities.

Reproduces Bernardi's formula for spanning forests of the hypercube.

Abstract

If $G$ is a strongly connected finite directed graph, the set $\mathcal{T}G$ of rooted directed spanning trees of $G$ is naturally equipped with a structure of directed graph: there is a directed edge from any spanning tree to any other obtained by adding an outgoing edge at its root vertex and deleting the outgoing edge of the endpoint. Any Schr\"odinger operator on $G$, for example the Laplacian, can be lifted canonically to $\mathcal{T}G$. We show that the determinant of such a lifted Schr\"odinger operator admits a remarkable factorization into a product of determinants of the restrictions of Schr\"odinger operators on subgraphs of $G$ and we give a combinatorial description of the multiplicities using an exploration procedure of the graph. A similar factorization can be obtained from earlier ideas of C. Athaniasadis, but this leads to a different expression of the multiplicities, as signed sums on which the nonnegativity is not appearent. We also provide a description of the block structure associated with this factorization. As a simple illustration we reprove a formula of Bernardi enumerating spanning forests of the hypercube, that is closely related to the graph of spanning trees of a bouquet. Several combinatorial questions are left open, such as giving a bijective interpretation of the results.