Bowen--Franks groups and minus class groups of cyclotomic number fields with prime conductor

Let $p$ be an odd rational prime and consider the cyclotomic number field $K=\mathbb{Q}({\zeta }_p)$ of conductor $p$. We construct a directed graph $Y$ on $p-1$ vertices for which the torsion part of the corresponding Bowen–Franks group is closely related to the minus part of the class group of $K$. In particular, both groups have the same cardinality up to an explicit power of $p$. Furthermore, they are both $\mathrm{G}\mathrm{a}\mathrm{l}(K\mathrm{/}\mathbb{Q})$-modules, and we prove the equality of the cardinalities of their isotypic components after tensoring them with the valuation ring of an appropriate $\mathrm{\ell }$-adic field for $\mathrm{\ell }\nmid p-1$.