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Inverse Fiedler vector problem of a graph

Jephian C. -H. Lin, Mahsa N Shirazi

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.

Inverse Fiedler vector problem of a graph

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.

Paper Structure

This paper contains 9 sections, 21 theorems, 43 equations, 7 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The inverse Fiedler vector problem of a tree
  4. Dirichlet matrices and its Perron vectors
  5. Weighted Laplacian matrices and Fiedler vectors
  6. Complete solutions
  7. Characteristic contraction and subdivision
  8. The inverse Fiedler vector problem of a cycle
  9. Concluding remark

Key Result

Theorem 2.2

Let $T$ be a tree. Then $\mathbf{x}$ is a Fiedler vector of $T$ if and only if $\mathbf{x}$ is Fiedler-like with respect to $T$.

Figures (7)

  • Figure 1: A weighted tree with a Type I Fiedler vector (marked in blue) and its characteristic set (marked in red).
  • Figure 2: A weighted tree with a Type II Fiedler vector (marked in blue) and its characteristic set (marked in red).
  • Figure 3: A tree with its boundary on a leaf $r = 1$
  • Figure 4: Three branches and their $\mathbf{x}_i$ (marked in blue) for Example \ref{['ex:t1treerecover']}, where on each edge the denominator records of $N_r^\top\mathbf{x}_i$ and the numerator records $N_r^{-1}\mathbf{x}_i$.
  • Figure 5: Two resistors in series.
  • ...and 2 more figures

Theorems & Definitions (51)

  • Example 1.2
  • Definition 2.1
  • Theorem 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8
  • ...and 41 more