The Kirchhoff's Matrix-Tree Theorem revisited: counting spanning trees with the quantum relative entropy

TL;DR

This paper revisits Kirchhoff's Matrix-Tree Theorem, providing an exact formula for counting spanning trees using quantum relative entropy and deriving bounds based on graph parameters.

Contribution

It introduces a novel quantum information perspective to count spanning trees and establishes tight bounds using quantum relative entropy.

Findings

Exact formula for spanning trees via quantum relative entropy

Tight bounds based on degrees and vertices

New quantum approach to classical graph enumeration

Abstract

By revisiting the Kirchhoff's Matrix-Tree Theorem, we give an exact formula for the number of spanning trees of a graph in terms of the quantum relative entropy between the maximally mixed state and another state specifically obtained from the graph. We use properties of the quantum relative entropy to prove tight bounds for the number of spanning trees in terms of basic parameters like degrees and number of vertices.