We provide eigenvalue asymptotics for a Dirac-type operator on Z(n) , n >= 2 , perturbed by multiplication operators that decay as |mu|(-gamma) with gamma < n . We show that the eigenvalues accumulate near the value of the flat band at a "semiclassical" rate with a constant that encodes the structure of the flat band. Similarly, we show that this behavior can be obtained also for a Laplace operator on a periodic graph.