A Graph Spectral Flow for Computing Nodal Deficiencies

TL;DR

This paper introduces a spectral flow method for graph Laplacians that accurately counts nodal domains of eigenvectors, extending continuous domain results to discrete graphs with numerical illustrations.

Contribution

It develops a novel spectral flow approach for graph Laplacians that links eigenvector nodal domains to spectral properties, extending prior continuous domain theories.

Findings

Spectral flow counts nodal domains on graphs.

The method extends continuous Laplacian results to discrete graphs.

Numerical examples validate the spectral flow approach.

Abstract

In this paper we propose a spectral flow for graph Laplacians, and prove that it counts the number of nodal domains for a given Laplace eigenvector. This extends work done for Laplacians on $\mathbb{R}^n$ to the graph setting. We mention some open problems relating the topology of a graph to the analytic behaviour of its Laplace eigenvectors, and include numerical examples illustrating our flow.