[Paper Review] Digraphs from Endomorphisms of Finite Cyclic Groups

Min Sha|arXiv (Cornell University)|Jul 10, 2010

graph theory and CDMA systems2 references3 citations

TL;DR

This paper introduces a family of directed graphs, G(n,k), constructed from endomorphisms of finite cyclic groups of order n, where each element maps to its k-th power. It characterizes the digraph's structure, including cycle lengths, automorphism groups, and adjacency matrix properties, showing that the automorphism group depends on the prime factorization of n and k, particularly when k is prime.

ABSTRACT

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

Motivation & Objective

• To define and analyze a new class of directed graphs G(n,k) derived from endomorphisms of finite cyclic groups.
• To characterize the structural properties of G(n,k), including cycle lengths, component decomposition, and vertex indegree distribution.
• To determine the automorphism group of G(n,k), especially when k is prime, and relate it to group actions and wreath products.
• To study the adjacency matrix and its characteristic and minimal polynomials, linking the results to monomial dynamical systems over finite fields.

Proposed method

• Construct G(n,k) as a directed graph with vertices as elements of a cyclic group H of order n, and a directed edge a → b if f(a) = b, where f(x) = x^k.
• Use number-theoretic tools such as greatest common divisors, least common multiples, and Euler's totient function to analyze solvability and preimage counts.
• Apply group-theoretic concepts including orders of elements, cyclic subgroups, and multiplicative order modulo d to classify cycle structures.
• Employ wreath product constructions to describe automorphism groups, particularly when k is prime and (n,k) = k.
• Use Lemma 2.1 to determine when a vertex has indegree > 0, based on the condition a^{n/d} = 1 with d = gcd(n,k).
• Leverage the fact that vertices in the same cycle have the same order and that cycle length ℓ(d) = ord_d(k), the multiplicative order of k modulo d.

Experimental results

Research questions

• RQ1 What is the structure of the digraph G(n,k) formed by the endomorphism f(x) = x^k on a finite cyclic group of order n?
• RQ2 How do the cycle lengths and number of cycles in G(n,k) depend on the parameters n and k?
• RQ3 What is the automorphism group of G(n,k), and how does it vary with the prime factorization of n and k?
• RQ4 How can the adjacency matrix of G(n,k) be characterized, and what are its characteristic and minimal polynomials?
• RQ5 Under what conditions is the automorphism group of the tree component T_1 isomorphic to a wreath product of symmetric and cyclic groups?

Key findings

• The number of vertices with indegree 0 in G(n,k) is (d-1)/d × n, where d = gcd(n,k), and vertices with positive indegree form a cyclic subgroup of size n/d.
• The number of cycle vertices in G(n,k) is exactly t, where t is the largest divisor of n coprime to k, and each cycle vertex has order dividing t.
• The length of a cycle with order d is ℓ(d) = ord_d(k), the multiplicative order of k modulo d, and the longest cycle has length ord_t(k).
• The total number of cycles in G(n,k) is ∑_{d|t} φ(d)/ℓ(d), where φ is Euler’s totient function and ℓ(d) = ord_d(k).
• When k is prime and gcd(n,k) = k, the automorphism group of the tree T_1 is isomorphic to Aut(T_1,h) ≅ Aut(T_1,h+1) ≀ S_k for h < h₀, with Aut(T_1,h₀) trivial.
• For k prime and gcd(n,k) = 1, the automorphism group of G(n,k) is isomorphic to the wreath product of symmetric groups over cyclic groups, specifically (Z_m ≀ S_{m_1}) × ⋯ × (Z_s ≀ S_{m_s})