  In this paper, continuous-time master equations with finite states employed
in nonequilibrium statistical mechanics are formulated in the language of
discrete geometry. In this formulation, chains in algebraic topology are used,
and master equations are described on graphs that consist of vertexes
representing states and of directed edges representing transition matrices. It
is then shown that master equations under the detailed balance conditions are
equivalent to discrete diffusion equations, where the Laplacians are defined as
self-adjoint operators with respect to introduced inner products. An
isospectral property of these Laplacians is shown for non-zero eigenvalues, and
its applications are given. The convergence to the equilibrium state is shown
by analyzing this class of diffusion equations. In addition, a systematic way
to derive closed dynamical systems for expectation values is given. For the
case that the detailed balance conditions are not imposed, master equations are
expressed as a form of a continuity equation.
