Dimension Bounds for Systems of Equations with Graph Structure

TL;DR

This paper introduces a graph-based framework for bounding the solution dimensions of systems of equations, leading to new bounds in spectral graph theory and nonlinear dynamics.

Contribution

It presents a general class of graph-structured equations and derives bounds on solutions using induced subgraphs, connecting to spectral and dynamical systems.

Findings

Bound on solution set dimension via induced subgraphs

New spectral bounds on graph eigenvalue multiplicities

Bounds on equilibrium set dimension in nonlinear networks

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.