Lifting of cycles in functional graphs

2025/09/16 by Tadahisa Nara, Nara, Tadahisa
Computer Science · #05C38 #11T06 #37P25 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.16234

openalex publication_date 2025/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given function from a set to itself, we can define a directed graph called the functional graph, where the vertices are the elements of the set, and the edges are all the pairs of inputs and outputs for the function. In this article we consider functional graphs on Z/mZ with respect to polynomial functions. The main result describes the behavior of cycles in functional graphs on Z/pⁿZ while n is increasing, where p is a prime number.

Citations

• On Functional Graphs of Quadratic Polynomials
• Functional Graphs of Polynomials over Finite Fields
• On the Number of Distinct Functional Graphs of Affine-Linear\n Transformations over Finite Fields

Related

• On Functional Graphs of Quadratic Polynomials
• Functional Graphs of Polynomials over Finite Fields
• Decomposition and factorisation of transients in Functional Graphs
• Graph structure of commuting functions
• The Graph Structure of Chebyshev Permutation Polynomials over Ring ℤ_p^k