In https://arxiv.org/pdf/1203.6133v3.pdf Kazunori Noguchi defines the zeta function for a finite category. It function is analogal to the Ihara zeta function of a graph. In http://emis.u-strasbg.fr/journals/EJC/Volume_16/PDF/v16i1r84.pdf says that there exist some cases when the ihara zet function determines the graph. My question is if there are a similar result about the Nouguchi's zeta function?.

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    $\begingroup$ In general, clearly no: the zeta function only depends on the underlying graph, in fact it only depends on the number of path of length $n$ in the underlying graph. For example if your category is just a groupe (groupoid with only one object) the zeta function only depends on the cardinal of the group. $\endgroup$ – Simon Henry Aug 10 '16 at 16:34

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