Complexity of Monad graphs generated by the function f(g) = g5

Abstract

A Monad graph is a graph Γ in which each of its vertices belongs to a finite group G and connects with its image under the action of a linear map f. This kind of graph was introduced by V. Arnold in 2003. In this paper, we compute the Monad graphs in which G is isomorphic to a cyclic group C_n of order n and f the fifth power function, i.e. f(g) = g⁵. Furthermore, some algebraic and dynamical properties of the studied Monad graphs are obtained. The proofs of our results are based on various tools and results with regard to the fields of number theory, algebra and graph theory.