\(\mathbb C\)-valued functions induced by graphs. (English) Zbl 1314.05228
Summary: In this paper, we establish certain analytic objects, \(\mathbb C\)-valued functions, containing combinatorial properties of given graphs. We call them graph (order-)zeta-functions. We study fundamental properties of such functions, and investigate algebras of the functions. Analytically, the convergence of them is considered. And we show that the construction of a system of graph zeta-functions is an invariance on finite connected graphs (up to shadowed graphs). Motivated by graph zeta-functions, we establish graph \(Z\)-functions, and the construction of them is an invariance on “locally finite” (finite or infinite) connected graphs (up to shadowed graphs).
MSC:
05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
11G15 | Complex multiplication and moduli of abelian varieties |
11R04 | Algebraic numbers; rings of algebraic integers |
11R09 | Polynomials (irreducibility, etc.) |
11R47 | Other analytic theory |
11R56 | Adèle rings and groups |
46L10 | General theory of von Neumann algebras |
46L40 | Automorphisms of selfadjoint operator algebras |
46L53 | Noncommutative probability and statistics |
46L54 | Free probability and free operator algebras |
References:
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