  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.
