Doubling constants and spectral theory on graphs

TL;DR

This paper investigates the minimal doubling constants on graphs, establishing a lower bound related to the spectral radius of the adjacency matrix, and characterizes graphs with small doubling constants, linking spectral and automorphism properties.

Contribution

It provides a lower bound for the doubling constant using spectral radius and characterizes graphs with small doubling constants, connecting spectral theory and graph automorphisms.

Findings

Lower bound for doubling constant: 1 + spectral radius of adjacency matrix

Complete characterization of graphs with doubling constant less than 3

Relation between automorphism group amenability and doubling minimizers

Abstract

We study the least doubling constant among all possible doubling measures defined on a (finite or infinite) graph $G$. We show that this constant can be estimated from below by $1+ r(A_G)$, where $r(A_G)$ is the spectral radius of the adjacency matrix of $G$, and study when both quantities coincide. We also illustrate how amenability of the automorphism group of a graph can be related to finding doubling minimizers. Finally, we give a complete characterization of graphs with doubling constant smaller than 3, in the spirit of Smith graphs.