  We find analytically the complete set of eigenvalues and eigenvectors
associated with Metropolis dynamics on a complete graph. As an application, we
use this information to study a counter-intuitive relaxation phenomenon, called
the Mpemba effect. This effect describes situations when upon performing a
thermal quench, a system prepared in equilibrium at high temperatures relaxes
faster to the bath temperature than a system prepared at a temperature closer
to that of the bath. We show that Metropolis dynamics on a complete graph does
not support weak nor strong Mpemba effect, however, when the graph is not
complete, the effect is possible.
