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{\bf R. Brown, I. Morris, J. Shrimpton and C.D. Wensley }
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{\bf Graphs of Morphisms of Graphs}
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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 We extend Shrimpton's (unpublished)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.
\bye