Spanning Tree Bounds for Grid Graphs

TL;DR

This paper establishes inequalities relating boundary size and the number of spanning trees in subgraphs of grid graphs, with applications to measuring district map compactness.

Contribution

It introduces new inequalities connecting boundary size and spanning tree counts in grid subgraphs, linking graph theory to district compactness measures.

Findings

Smaller boundary subgraphs have more spanning trees.

Larger boundary subgraphs have fewer spanning trees.

Application to district map compactness metrics.

Abstract

Among subgraphs with a fixed number of vertices of the regular square lattice, we prove inequalities that essentially say that those with smaller boundaries have larger numbers of spanning trees and vice-versa. As an application, we relate two commonly used measurements of the compactness of district maps.