We consider compact manifolds $M$ with a cohomogeneity one action of Lie group $G$ such that the orbit space $M/G$ is closed interval. For $T$ maximal torus $G$, we find necessary and sufficient conditions on diagram $T$-action GKM type, describe its graph. The general results are illustrated explicit examples.

We investigate the geometry of Hermitian manifolds endowed with a compact Lie group action by holomorphic isometries principal orbits codimension one. In particular, we focus on special class these constructed following B\'erard-Bergery which includes, among others, line bundles $\mathbb C\mathbb P^{m-1}$, linear Hopf and Hirzebruch surfaces. characterize their invariant metrics, such as balanc...