  The electronic structures of two mixed valence insulators YbB$_6$ and
YbB$_12$ are studied by using the local density approximation (LDA)
supplemented with the Gutzwiller method and dynamic mean field theory (DMFT).
YbB$_6$ is found to be a moderately correlated Z$_2$ topological insulator,
similar to SmB$_6$ but having much larger bulk band gap. Notably, YbB$_12$ is
revealed to be in a new novel quantum state, strongly correlated topological
crystalline Kondo insulator, which is characterized by its non-zero mirror
Chern number. The surface calculations find odd (three) and even (four) number
of Dirac cones for YbB$_6$ and YbB$_12$, respectively.
