Universal Properties of Some Quivers
  In this paper, I characterize four particular classes of directed multigraphs, or quivers, as images under left and right adjoints to the natural vertex and edge functors. In particular, the following notions coincide: (1) independent sets of vertices with a left adjoint functor to the vertex functor, (2) independent sets of edges with a left adjoint functor to the edge functor, (3) complete digraphs with a right adjoint functor to the vertex functor, (4) bouquets with a right adjoint functor to the edge functor. 
Will Grilliette
math.CT math.CO