Cayley graphs and endomorphism monoids

Abstract

Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Curs: 2022-2023. Director: Kolja Knauer[en] Since its first steps at the hands of Euler, graph theory has gradually become a field of great interest and innovation for the mathematical community. From its surprising capability to simplify the formulation of applied problems to the rich complexity that some of its natural problems contain, even in finite settings (being the field to see the first computer-assisted proof in mathematics), the list of its merits and uses seems to only grow in length, keeping the promise of attracting research for the times to come. For the untrained eye, however, it could appear as a branch with few theoretically rich connections to other fields of mathematics aside from topology (via graph embeddings), which is a misconception. In a certain way, disproving this thought is the main focus of this project, as the aim is to show the connections between graph theory and abstract algebra (semigroup and monoid theory specifically), hoping to put both in a more interesting light. Specifically, we introduce and talk about the basic tools of the field (mainly the Cayley graph construction, and a fairly young generalization of it by Yongwen Zhu as seen in [10]), introduce some recent interesting results in the literature by many authors, mostly by K.Knauer and coauthors (as in references [5], [7], [6]) and try to put together a comprehensive guide to try and understand the main difficulties and ideas used in one of the main lines of work in the field. This can be exemplified in our in-depth study of some families of outerplanar graphs as monoid graphs, or our brief study of $K_4 \sqcup C_5$ as a non monoid but possibly semigroup graph. Both questions were originally raised by K.Knauer and Puig i Surroca in their work referenced in [5]