Opuscula Math. 37, no. 2 (2017), 225-263
http://dx.doi.org/10.7494/OpMath.2017.37.2.225

Opuscula Mathematica

# Non-factorizable C-valued functions induced by finite connected graphs

Ilwoo Cho

Abstract. In this paper, we study factorizability of $$\mathbb{C}$$-valued formal series at fixed vertices, called the graph zeta functions, induced by the reduced length on the graph groupoids of given finite connected directed graphs. The construction of such functions is motivated by that of Redei zeta functions. In particular, we are interested in (i) "non-factorizability" of such functions, and (ii) certain factorizable functions induced by non-factorizable functions. By constructing factorizable functions from our non-factorizable functions, we study relations between graph zeta functions and well-known number-theoretic objects, the Riemann zeta function and the Euler totient function.

Keywords: directed graphs, graph groupoids, Redei zeta functions, graph zeta functions, non-factorizable graphs, gluing on graphs.

Mathematics Subject Classification: 05E15, 11G15, 11R47, 11R56, 46L10, 46L40, 46L54.

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• Ilwoo Cho
• St. Ambrose University, Department of Mathematics and Statistics, 421 Ambrose Hall, 518 W. Locust St., Davenport, Iowa, 52803, USA
• Communicated by P.A. Cojuhari.
• Revised: 2016-07-21.
• Accepted: 2016-08-13.
• Published online: 2017-01-03.