# Invertibility of graph translation and support of Laplacian Fiedler vectors

## TL;DR

This paper investigates the invertibility of graph translation operators based on Laplacian eigenvectors and explores the support properties of Fiedler vectors in planar and non-planar graphs.

## Contribution

It characterizes when graph translation is invertible in terms of eigenvector support and analyzes the support of Fiedler vectors in different graph classes.

## Key findings

- Translation operator invertibility depends on eigenvector support.
- Fiedler vector of planar graphs cannot vanish on large neighborhoods.
- Constructed non-planar graphs with vanishing Fiedler vectors.

## Abstract

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the $i$'th node is invertible if and only if all eigenvectors are nonzero on the $i$'th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property.