Comparing Graphs of Different Sizes

TL;DR

This paper introduces methods to compare finite graphs of different sizes using notions of size, and proves theorems relating their combinatorial properties such as spanning trees and random walk probabilities.

Contribution

It develops new inequalities and theoretical tools for analyzing and comparing graphs of different sizes based on their structural and probabilistic properties.

Findings

Theorems relating the number of spanning trees between graphs of different sizes.

Results connecting return probabilities of random walks to graph size.

Inequalities involving determinants, traces, and entropy for graph comparison.

Abstract

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.