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A linear system $\dot x = Ax$, $A \in \mathbb{R}^{n \times n}$, $x \in\mathbb{R}^n$, with $\mathrm{rk} A = n-1$, has a one-dimensional centermanifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. If a differential equation$\dot x = f(x)$ 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 $\mathbb{C}^+$, 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 formulatemeso-scale conditions with certain principal minors of $A$ which are necessaryfor semi-definiteness. We illustrate these results by the example of theKuramoto model of coupled oscillators.

Meso-scale obstructions to stability of 1D center manifolds for networks of coupled differential equations with symmetric Jacobian