  We show the existence of invariant energy levels in a Kondo lattice model on
an isolated complete graph, such as a triangle and a tetrahedron. These energy
levels always have fixed eigenenergies $t \pm J/2$, irrespective of the
configuration of localized moments ($t$ is the transfer integral of conduction
electrons and $J$ is the spin-charge coupling constant). We also extend the
analysis to geometrically frustrated lattices by using the complete graphs as
basic building blocks. We show that the construction rule for the invariant
energy levels leads to the existence condition of localized states, if the
model is defined on the triangle-based line graphs, such as a kagome lattice.
We further propose a procedure of engineering isolated flat bands with broken
time-reversal symmetry, which are separated from other dispersive bands with
finite energy gaps.
