\(\mathbb C\)-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:
| [1] | Cho, I.: Operators induced by prime numbers. Methods Appl. Math. Sci. (2013, to appear) · Zbl 1332.46064 |
| [2] | Cho, I.: Classification on arithmetic functions and corresponding free-moment \[L\] L-functions. Bull. Korea Math. Soc. (2013, to appear) · Zbl 1329.11113 |
| [3] | Cho, \[I.: p\] p-Adic Banach-space operators and adelic Banach-space operators. Opusc. Math. (2013, to appear) · Zbl 1428.47032 |
| [4] | Cho, I., Gillespie, T.: Arithmetic functions and corresponding free probability determined by primes. Rocky Mt. J. Math. (2013, submitted) · Zbl 1295.05160 |
| [5] | Cho, I.: Graph Groupoids and Partial Isometries. Lambert Academic Press, Saarbrücken (2009). ISBN: 978-3-8383-1397-9 · Zbl 0446.05003 |
| [6] | Cho, I.: Fractals on Graphs. Verlag with Dr. Muller (2009). ISBN: 978-3-639-19447-0 |
| [7] | Cho, I.: Operations on Graphs, Groupoids, and Operator Algebras. Lambert Academic Press, Saarbrücken (2010). ISBN: 978-8383-5271-8 |
| [8] | Cho, I., Jorgensen, P.E.T.: Operators induced by graphs. Lett. Math. Phys. (2012, to appear). doi:10.1007/s11005-012-0575-4 · Zbl 1273.05149 |
| [9] | Cho, I., Jorgensen, P.E.T.: Moment computations of graphs with fractal property. J. Appl. Math. Comput. 37, 377-406 (2011) · Zbl 1295.05160 · doi:10.1007/s12190-010-0440-5 |
| [10] | Cho, I.: Histories distorted by partial isometries. J. Phys. Math. (2012, to appear) · Zbl 1264.81207 |
| [11] | Gillespie, T.: Prime number theorems for Rankin-Selberg \[L\] L-functions over number fields. Sci. China Math. 54(1), 35-46 (2011) · Zbl 1251.11035 · doi:10.1007/s11425-010-4137-x |
| [12] | Bump, D.: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics, vol. 55. Cambridge University Press, Cambridge (1996). ISBN: 0-521-65818-7 |
| [13] | Kung, J.P.S., Murty, M.R., Rota, G.-C.: On the Rédei zeta function. J. Number Theory 12, 421-436 (1980) · Zbl 0446.05003 · doi:10.1016/0022-314X(80)90034-7 |
| [14] | Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009). ISBN: 978-0-521-89806-5 · Zbl 1165.05001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
