Incidence graphs and non-negative integral quadratic forms

We present directed multigraphs (quivers) and their walks as combinatorial models for non-negative semi-unit integral quadratic forms of Dynkin type A and their associated roots and radical vectors. This point of view leads to an easy description of those forms which are positive and weakly positive (properties of interest in Lie theory and the representation theory of algebras), and derives into a categorification of such unitary forms.