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We apply single and double tree-like representations of Markov jump processeson $\mathbb{Z}_N$ for obtaining their nonequilibrium heat capacity and fortaking the diffusion limit $N\uparrow \infty$. The main tool is a graphicalrepresentation of the Drazin-inverse of the backward generator. In that way,the combination of algebraic and graph-theoretical approaches enables an exactcomputation of an important nonequilibrium quantity (excess heat) for Markovjump processes.

Drazin-Inverse and heat capacity for driven random walks on the ring