On Quotient Spaces of Compact Lie Groups by Tori Centralizers

We consider the graph of the homogeneous space $K/L$, where $K$ is a compact Lie group and $L$ is the centralizer of a torus in $K$. We obtain a characterization of those spaces whose graphs admit embeddings in a certain standard graph. We compute the number of arcs in such graphs. We also give a simple expression for the Euler class of the homogeneous space $K/L$.