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Stability of critical equipartitions of graphs

Connor Menzel

Abstract

We consider partitions of a finite, simple, weighted graph that minimize a spectral energy functional, defined to be the maximum of the first eigenvalues on each component. These partitions are minimized with respect to a parameter that we view as a small perturbation of a fixed combinatorial partition. It has been shown that critical points of the energy functional in this framework correspond to non-degenerate eigenvectors of the graph Laplacian if and only if the partition is bipartite. In this work we generalize this result to partitions that are not necessarily bipartite by constructing a modified graph Laplacian called the partition Laplacian. The main result states that critical points of the spectral energy functional correspond to nodal partitions of non-degenerate eigenvectors of the partition Laplacian. Furthermore, the stability of each critical point is determined entirely by the nodal deficiency of the associated eigenvector. We also consider more general signed Laplacians and prove that Courant-sharp Laplacian eigenvectors induce globally minimal partitions.

Stability of critical equipartitions of graphs

Abstract

We consider partitions of a finite, simple, weighted graph that minimize a spectral energy functional, defined to be the maximum of the first eigenvalues on each component. These partitions are minimized with respect to a parameter that we view as a small perturbation of a fixed combinatorial partition. It has been shown that critical points of the energy functional in this framework correspond to non-degenerate eigenvectors of the graph Laplacian if and only if the partition is bipartite. In this work we generalize this result to partitions that are not necessarily bipartite by constructing a modified graph Laplacian called the partition Laplacian. The main result states that critical points of the spectral energy functional correspond to nodal partitions of non-degenerate eigenvectors of the partition Laplacian. Furthermore, the stability of each critical point is determined entirely by the nodal deficiency of the associated eigenvector. We also consider more general signed Laplacians and prove that Courant-sharp Laplacian eigenvectors induce globally minimal partitions.
Paper Structure (18 sections, 18 theorems, 82 equations, 5 figures)

This paper contains 18 sections, 18 theorems, 82 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Spectral minimal partitions
  3. Outline
  4. Acknowledgments
  5. Definitions and Background
  6. Signed graphs
  7. Nodal domains on signed graphs
  8. Graph partitions and the partition Laplacian
  9. Parameter-Dependent Partitions
  10. Removal of a single edge
  11. Upgrading to partitions
  12. Proof of Theorem \ref{['thm:main_result']}
  13. Switching Equivalence and the Partition Energy
  14. Connection to the partition Laplacian on surfaces
  15. The continuum partition Laplacian
  16. ...and 3 more sections

Key Result

Theorem 1.1

Given a partition $P$ of a graph $G$ into $\nu$ connected subgraphs $\{G_k\}_{k=1}^{\nu}$, consider local perturbations of $P$ parametrized by $\alpha$. Define On a smooth submanifold of the parameter space, there is a one-to-one correspondence between critical points of the map $\alpha \mapsto \Lambda(P,\alpha)$ and eigenvectors of the partition Laplacian $L^{\partial P}$ with nodal partition $P

Figures (5)

  • Figure 1: The nodal diagram for an eigenvector corresponding to the second eigenvalue on the star graph with 5 vertices. This eigenvector has 4 strong nodal domains but only two weak nodal domains.
  • Figure 2: The 3-cycle graph $C_3$ with three distinct signatures. (A) and (B) are switching equivalent, but (C) is not switching equivalent to either because the sign of the cycle in (C) is negative.
  • Figure 3: A simple graph $G$ with desired connected components $G_k$ of 3-partition $P$ circled, the resultant 3-partition $P$, and the associated multigraph $P^G$.
  • Figure 4: A schematic of the transversality argument with $|\partial P| = \nu = 2$. The matrix $d\Phi_{\alpha}$ may not have full rank at every $\alpha$, but the key is that the tangent space of $\Delta$ always complements the image of $d\Phi_{\alpha}$ to give all of $\mathbb{R}^{\nu}$.
  • Figure 5: The base graph $G$ on the left with some critical equipartition $(P,\alpha^c)$. This is the nodal partition of some eigenvector $\psi$ of $L^{\partial P}$. By removing one edge, $\psi$ induces a critical equipartition of the modified graph $G_{\xi}$. This is a tree partition, and is hence unique, so $\psi$ must be Courant-sharp as an eigenvector on $G_{\xi}$.

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1: zaslavskySignedGraphs1982
  • Lemma 2.2: hararyNotionBalanceSigned1953
  • Remark 2.1
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.3: geNodalDomainTheorems2023
  • ...and 28 more