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A note on Puder's generalised co-growth formula for trees

Wenbo Li, Joe Thomas

TL;DR

The paper proves Puder's generalised co-growth formula for nonnegative functions on bi-regular trees by developing an operator-level resolvent identity that connects the adjacency operator to non-backtracking walk matrices. The authors introduce the bi-resolvent, derive its expansion on bi-regular trees, and leverage the Hashimoto framework to link growth rates $\alpha(f)$ and $\beta(f)$, establishing the explicit threshold-based formulas for bi-regular trees and providing a streamlined proof for regular trees. The results lift to general bi-regular graphs, preserving the growth rates, and have potential implications for understanding spectral outliers and staircase phenomena in random regular/bi-regular graphs. The approach mirrors modern resolvent-based proofs of co-growth, adapting them to the bi-regular setting to yield exact growth-rate characterisations.

Abstract

In this note, we prove a conjecture of Puder on an extension of the co-growth formula to any non-negative function defined on a bi-regular tree. A key component of our proof is the establishment of a resolvent identity, which serves as an operator version of the co-growth formula. We also provide a simpler proof of Puder's generalised co-growth formula for the regular tree.

A note on Puder's generalised co-growth formula for trees

TL;DR

The paper proves Puder's generalised co-growth formula for nonnegative functions on bi-regular trees by developing an operator-level resolvent identity that connects the adjacency operator to non-backtracking walk matrices. The authors introduce the bi-resolvent, derive its expansion on bi-regular trees, and leverage the Hashimoto framework to link growth rates and , establishing the explicit threshold-based formulas for bi-regular trees and providing a streamlined proof for regular trees. The results lift to general bi-regular graphs, preserving the growth rates, and have potential implications for understanding spectral outliers and staircase phenomena in random regular/bi-regular graphs. The approach mirrors modern resolvent-based proofs of co-growth, adapting them to the bi-regular setting to yield exact growth-rate characterisations.

Abstract

In this note, we prove a conjecture of Puder on an extension of the co-growth formula to any non-negative function defined on a bi-regular tree. A key component of our proof is the establishment of a resolvent identity, which serves as an operator version of the co-growth formula. We also provide a simpler proof of Puder's generalised co-growth formula for the regular tree.

Paper Structure

This paper contains 4 sections, 6 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. The regular tree
  3. The bi-regular tree
  4. Vertex independence for $\alpha(f)$ and $\beta(f)$

Key Result

Theorem 1.1

Let $f\neq 0$ be a non-negative function on a $d$-regular tree with $d\geq 3$. Then, Moreover, if $f=\mathds{1}_S$ for $\emptyset\neq S\subseteq V(G)$ and $G$ is a $(k,l)$-bi-regular tree, then

Theorems & Definitions (13)

  • Theorem 1.1: Puder2024
  • Theorem 1.2
  • Corollary 1.1
  • proof
  • proof : Proof of Theorem \ref{['thm:co-growth']} for the regular tree
  • Remark 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 3 more