Zeta functions of finite graphs and coverings. II.

*(English)*Zbl 0972.11086Part I, cf. Adv. Math. 121, 124-165 (1996; Zbl 0874.11064).

This is the second part of the authors’ paper on zeta functions associated to finite graphs. They developed a Galois theory for normal unramified coverings of finite irregular graphs and established analogues between the graph zeta-functions and the classical Artin’s \(L\)-functions. They defined three kinds of \(L\)-functions attached to finite graphs as an analogy of Artin’s \(L\)-functions for algebraic number fields. These three types are based on vertex variables, edge variables, and path variables.

Analogues of all the standard Artin’s \(L\)-functions results for number fields are proved here for all three types of \(L\)-functions. In particular, they obtain factorization formulas for the zeta functions introduced in Part I as a product of \(L\)-functions. It is shown that the path \(L\)-functions, which depend only on the rank of the graph, can be specialized to give the edge \(L\)-functions, and these in turn can be specialized to give the vertex \(L\)-functions. In order to prove that Ihara type quadratic formulas hold for vertex \(L\)-functions, the method of Bass is used. Finally, they use the theory to give examples of two regular graphs without multiple edges or loops having the same vertex zeta functions. These graphs are also isospectral but not isomorphic.

This is the second part of the authors’ paper on zeta functions associated to finite graphs. They developed a Galois theory for normal unramified coverings of finite irregular graphs and established analogues between the graph zeta-functions and the classical Artin’s \(L\)-functions. They defined three kinds of \(L\)-functions attached to finite graphs as an analogy of Artin’s \(L\)-functions for algebraic number fields. These three types are based on vertex variables, edge variables, and path variables.

Analogues of all the standard Artin’s \(L\)-functions results for number fields are proved here for all three types of \(L\)-functions. In particular, they obtain factorization formulas for the zeta functions introduced in Part I as a product of \(L\)-functions. It is shown that the path \(L\)-functions, which depend only on the rank of the graph, can be specialized to give the edge \(L\)-functions, and these in turn can be specialized to give the vertex \(L\)-functions. In order to prove that Ihara type quadratic formulas hold for vertex \(L\)-functions, the method of Bass is used. Finally, they use the theory to give examples of two regular graphs without multiple edges or loops having the same vertex zeta functions. These graphs are also isospectral but not isomorphic.

Reviewer: Masakazu Muro (Yanagido)

##### MSC:

11M41 | Other Dirichlet series and zeta functions |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

11T99 | Finite fields and commutative rings (number-theoretic aspects) |

##### Keywords:

Ihara type quadratic formulas; zeta functions associated to finite graphs; \(L\)-functions attached to finite graphs; vertex variables; edge variables; path variables; factorization formulas; product of \(L\)-functions; path \(L\)-functions; edge \(L\)-functions; vertex \(L\)-functions; Ihara type
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\textit{H. M. Stark} and \textit{A. A. Terras}, Adv. Math. 154, No. 1, 132--195 (2000; Zbl 0972.11086)

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