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Remarks on the zeta function of a graph. (English) Zbl 1174.11391

Summary: We make two observations about the zeta function of a graph. First we show how Bass’s proof of Ihara’s formula fits into the framework of torsion of complexes. Second, we show how in the special case of those graphs that are quotients of the Bruhat-Tits tree for \(\text{SL}(2; K)\) for a local nonarchimedean field \(K\), the zeta function has a natural expression in terms of the \(L\)-functions of Coxeter systems.

MSC:

11M41 Other Dirichlet series and zeta functions
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
14G35 Modular and Shimura varieties
20E42 Groups with a \(BN\)-pair; buildings