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Dimension Bounds for Systems of Equations with Graph Structure

Eddie Nijholt, Davide Sclosa

Abstract

We introduce a broad class of equations that are described by a graph, which includes many well-studied systems. For these, we show that the number of solutions (or the dimension of the solution set) can be bounded by studying certain induced subgraphs. As corollaries, we obtain novel bounds in spectral graph theory on the multiplicities of graph eigenvalues, and in nonlinear dynamical system on the dimension of the equilibrium set of a network.

Dimension Bounds for Systems of Equations with Graph Structure

Abstract

We introduce a broad class of equations that are described by a graph, which includes many well-studied systems. For these, we show that the number of solutions (or the dimension of the solution set) can be bounded by studying certain induced subgraphs. As corollaries, we obtain novel bounds in spectral graph theory on the multiplicities of graph eigenvalues, and in nonlinear dynamical system on the dimension of the equilibrium set of a network.

Paper Structure

This paper contains 9 sections, 16 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Finite Determinacy, Compatibility, and Combinatorial Dimension
  3. Finite Determinacy
  4. Equations Compatible with a Graph
  5. Combinatorial dimension
  6. Main Theorem
  7. Applications to Spectral Graph Theory
  8. Applications to Dynamical Systems
  9. Acknowledgments

Key Result

Theorem 1

Consider a system of equations in the variables $(x_v)_{v\in V}$ that is "sufficiently compatible" with a graph $G=(V,E)$, in a way that will be made precise. Let $F\subseteq G$ be an induced forest. Let $L\subseteq F$ be obtained by choosing from each component of $F$ all leaves except one. If valu

Figures (3)

  • Figure 1: An induced subforest (black) of a graph (gray) and a set of selected leaves (squares) obtained by choosing from each component of the forest all leaves except one.
  • Figure 2: The star graph with $n=10$ vertices, together with two examples of an induced subforest (black) with selected leaves (squares). The subforest on the left gives a bound of $8$ on the multiplicity of any eigenvalue (in general, $n-2$). The subforest on the right gives a bound of $1$, but does not apply to all eigenvalues due to isolated vertices.
  • Figure 3: A tree is obtained from a set of cycle subgraphs. Any two of these cycles have common edges if and only if they are adjacent in the tree, and if they do they have a single common edge. In contrast, there is no restriction on the common vertices.

Theorems & Definitions (44)

  • Theorem : Forest Bound, Informal
  • Definition 2
  • Example 3
  • Remark 4
  • Remark 5
  • Lemma 6
  • proof
  • Definition 7
  • Definition 8
  • Example 9: Graph Spectrum
  • ...and 34 more