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.