Inverse Fiedler vector problem of a graph

TL;DR

This paper investigates how the structure of a graph influences its Fiedler vectors, characterizing all possible eigenvectors for trees and cycles, which has implications for spectral graph analysis.

Contribution

It provides a complete characterization of all possible Fiedler vectors for trees and cycles based on their structure, advancing understanding of spectral properties.

Findings

Characterization of all Fiedler vectors for trees

Characterization of eigenvectors for cycles

Implications for graph partitioning and spectral clustering

Abstract

Given a graph and one of its weighted Laplacian matrix, a Fiedler vector is an eigenvector with respect to the second smallest eigenvalue. The Fiedler vectors have been used widely for graph partitioning, graph drawing, spectral clustering, and finding the characteristic set. This paper studies how the graph structure can control the possible Fiedler vectors for different weighted Laplacian matrices. For a given tree, we characterize all possible Fiedler vectors among its weighted Laplacian matrix. As an application, the characteristic set can be anywhere on a tree, except for the set containing a single leaf. For a given cycle, we characterize all possible eigenvectors corresponding to the second or the third smallest eigenvalue.