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 |