Laplacians on infinite graphs: discrete vs continuous

TL;DR

This paper explores the relationship between discrete graph Laplacians and continuous Laplacians on metric graphs, emphasizing their complementary nature and how their interplay enhances understanding in mathematics and physics.

Contribution

It provides a comprehensive overview of the spectral and parabolic properties linking discrete and continuous Laplacians on graphs, highlighting their interconnected roles.

Findings

Discrete and continuous Laplacians are complementary, not opposing.

Interplay between the two offers deeper insights into spectral properties.

Unified perspective benefits mathematical physics and graph theory.

Abstract

There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this overview, we will focus on the relationship between them (spectral and parabolic properties). Our main conceptual message is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides.