On monoids of endomorphisms of a cycle graph

TL;DR

This paper studies the structure of monoids of endomorphisms of cycle graphs, providing methods to find minimal generating sets, describing algebraic relations, and calculating their sizes.

Contribution

It introduces a process to determine minimal generators for the monoids of endomorphisms and weak endomorphisms of cycle graphs, and analyzes their algebraic properties.

Findings

Determined minimal generating sets for the monoids.

Described Green's relations and regularity properties.

Calculated the cardinalities of the monoids.

Abstract

In this paper we consider endomorphisms of an undirected cycle graph from Semigroup Theory perspective. Our main aim is to present a process to determine sets of generators with minimal cardinality for the monoids $wEnd(C_n)$ and $End(C_n)$ of all weak endomorphisms and all endomorphisms of an undirected cycle graph $C_n$ with $n$ vertices. We also describe Green's relations and regularity of these monoids and calculate their cardinalities.