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

Graph-combinatorial approach for large deviations of Markov chains