${\mathbb{F}}_p$ and $Z_p$ Valued Holomorphic Functions over Graphs

The definition of a holomorphic function over a general measurable space $S$ endowed with a Markov process is defined by Zeghib and Barre. In this article we consider holomorphic functions over graphs whose ranges are a given finite field or a cyclic group. Also we consider a relation between $\mathbb{C}$-holomorphic functions over regular trees and the field of $p$-adic numbers.