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}
}
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 We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case… 

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