2008/04/03 by Ronnie Brown, I. Morris, John Shrimpton +1 · 1 citation
Mathematics · Computer Science · #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #Algebraic structures and combinatorial models #Morphism #Mathematics #Combinatorics #Undirected graph #Monoid #Endomorphism #Discrete mathematics #Vertex (graph theory) #Equivalence (formal languages) #Directed graph #Graph
paper · pdf · doi:10.37236/919
openalex publication_date 2008/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/26
This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.