  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 center
manifold $E^c = \{v \in \mathbb{R}^n : Av=0\}$. 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 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.
