Some groups related to the symmetry of a directed graph

Abstract

This paper studies a new approach to analyzing the symmetry of a graph, using the idea of a ‘symmetry object’: this is a generalization of an automorphism group, which possesses the structure of an object in the category under consideration. To define a symmetry object we use categorical methods, and in particular the notions of cartesian closed structure and monoidal closed structure. A feature of cartesian closed categories is that they permit the construction of symmetry objects which are group objects in the category. We use this property to pursue some analogies with group theory in a category of directed graphs. We associate a crossed module μ: Q( Γ)→Aut( Γ) to the automorphism group of a directed graph Γ, by showing that Aut( Γ) can be embedded in the symmetry object which is both a group and a directed graph. By comparing this crossed module with the corresponding construction for groups (considered as groupoids, to give a cartesian closed category) we define inner automorphism, outer automorphism and centre groups for Γ. We show that these can be completely described in terms of the way that the automorphisms act on a particular class of subgraphs of Γ.