Independence of homogeneous GKM manifolds and symmetric spaces

Let $G\mathrm{/}H$ be a simply connected homogeneous space of maximal rank. Then the maximal torus $T$-action on $G\mathrm{/}H$ is a GKM manifold. We call the $T$-action $j$-independent if any $i(\le j)$ pairwise distinct isotropy weights at a fixed point are linearly independent. Using weighted graphs, we show that the maximal independence of $G\mathrm{/}H$ is $2$, $3$ or $n=\dim T$, and that the cases of $3$ or $n=\dim T$ correspond to some symmetric spaces of rank $>2$. As a corollary, using the results of Ayzenberg and Masuda, the lower-degree reduced homology groups (with appropriate coefficients) of the orbit space $T\mathrm{\backslash }G\mathrm{/}H$ vanish.