Graphs of Morphisms of Graphs

@article{Brown2008GraphsOM,
title={Graphs of Morphisms of Graphs},
author={Ronald Brown and Ifor Morris and John S. Shrimpton and Christopher D. Wensley},
journal={Electron. J. Comb.},
year={2008},
volume={15}
}
• Published 3 April 2008
• Mathematics
• Electron. J. Comb.
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 We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case…
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References

SHOWING 1-10 OF 33 REFERENCES

Algebraic structures on graphs

In this paper we consider directed graphs with algebraic structures: group-graphs, ringgraphs, involutorial graphs, affine graphs, graphs of morphisms between graphs, graphs of reduced paths of an

Higher order symmetry of graphs

• Ronald Brown
• Mathematics
Irish Mathematical Society Bulletin
• 1994
1 Symmetry in analogues of set theory This lecture gives background to and results of work of my student John Shrimpton [19, 20, 21]. It advertises the joining of two themes: groups and symmetry; and

Topology of finite graphs

This paper derives from a course in group theory which I gave at Berkeley in 1982. I wanted to prove the standard theorems on free groups, and discovered that, after a few preliminaries, the notion

AN INTRODUCTION TO THE CATEGORY OF GRAPHS *

In recent years the category-theoretical approach to graph theory has gained considerably in acceptance, mainly because of its most spectacular successes [41, 47, 481. It is our intention here to

Fibrations of graphs

• Mathematics
Discret. Math.
• 2002

Handbook of Categorical Algebra

The Handbook of Categorical Algebra is intended to give, in three volumes, a rather detailed account of what, ideally, everybody working in category theory should know, whatever the specific topic of

Groupoids in combinatorics -- applications of a theory of local symmetries

An objective of the theory of combinatorial groupoids is to introduce concepts like "holonomy", "parallel transport", "bundles", "combinatorial curvature" etc. in the context of simplicial

Sheaves in geometry and logic: a first introduction to topos theory

• Mathematics
• 1992
This text presents topos theory as it has developed from the study of sheaves. Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various

A Guided Tour in the Topos of Graphs

In this paper we survey the fundamental constructions of a presheaf topos in the case of the elementary topos of graphs. We prove that the transition graphs of nondeterministic automata (a.k.a.