A linear system ẋ = Ax, A ∈ ℝ^n×n, x ∈ ℝⁿ, with rkA = n - 1, has a one-dimensional center manifold E^c = {v ∈ ℝⁿ : Av = 0}. If a differential equation ẋ = f (x) has a one-dimensional center manifold W^c at an equilibrium x^* then E^c is tangential to W ^c with A = Df (x^*) and for stability of W ^c it is necessary that A has no spectrum in C⁺, i.e. if A is symmetric, it has to be negative semi-definite. We establish a graph theoretical approach to characterize semi-definiteness. Using spanning trees for the graph corresponding to A, we formulate meso-scale conditions with certain principal minors of A which are necessary for semi-definiteness. We illustrate these results by the example of the Kuramoto model of coupled oscillators.