# Graph properties, categorically defined

Consider the category of finite graphs with graph homomorphisms as morphisms.

Are there interesting graph properties that can be defined in categorical language? Can for example connectedness be defined in categorical language?

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Disjoint union is the coproduct in the category of finite graphs, so connected graphs are precisely the noninitial objects in this category that can not be expressed as a coproduct of two nonempty subobjects. See the entry on connected object in the nlab.

If you want to read more about arguments that intertwine classical results in graphs theory with category theory I suggest the book "Graphs and homomorphisms" by J. Nesetril.

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The subcategory of graphs without loops is not that well behaved from the point of view of category theory, But graphs without loops have their merits too: A graph morphism from $(V,E)$ to the complete graph without loops on $n$ vertices is the same thing as a $n$-vertex-coloring of $(V,E)$.