Graphs, spectral triples and Dirac zeta functions

To a finite, connected, unoriented graph of Betti-number g ≥ 2 and valencies ≥ 3 we associate a finitely summable, commutative spectral triple (in the sense of Connes), whose induced zeta functions encode the graph. This gives another example where non-commutative geometry provides a rigid framework for classification.