Approximately uniformly locally finite graphs

TL;DR

This paper investigates a class of locally finite graphs where the Laplacian matrix can be approximated by matrices with bounded non-zero entries per row and column, revealing regularity constraints related to vertex degrees.

Contribution

It introduces and characterizes a new class of locally finite graphs with Laplacian matrices approximable by bounded-sparsity matrices, highlighting their regularity properties.

Findings

Graphs in this class cannot have high-degree vertices connected to many low-degree vertices.

Examples of graphs both within and outside this class are provided.

The class exhibits a regularity constraint linking vertex degrees.

Abstract

Let $\Gamma$ be a locally finite graph, $L$ the normalized Laplacian of $\Gamma$. If $\Gamma$ is uniformy locally finite, i.e. if each vertex has no more than $d$ adjacent vertices, then the matrix of $L$ (with respect to the standard basis) has no more than $d+1$ non-zero entries in each row and in each column. We consider the class of locally finite graphs, for which the Laplacian can be approximated by matrices of this type with arbitrary $d$. We provide examples of locally finite graphs which are or are not in this class, and show that the graphs from this class share certain regularity property: vertices of high degree cannot have too many adjacent vertices of low degree.