  We consider discrete-time Markov chains and study large deviations of the
pair empirical occupation measure, which is useful to compute fluctuations of
pure-additive and jump-type observables. We provide an exact expression for the
finite-time moment generating function, which is split in cycles and paths
contributions, and scaled cumulant generating function of the pair empirical
occupation measure via a graph-combinatorial approach. The expression obtained
allows us to give a physical interpretation of interaction and entropic terms,
and of the Lagrange multipliers, and may serve as a starting point for
sub-leading asymptotics. We illustrate the use of the method for a simple
two-state Markov chain.
