In this paper, we study the incidence algebra I (A, γ) of the category of paths in a (oriented multi-) graph γ over a ring A.We describe relationships between algebraic properties of I(A, We describe relationships between algebraic properties of I(A, γ) and combinatorial features of γ. Specifically, we give necessary and sufficient conditions on γ and A in order that I(A, γ) satisfy a polynomial identity, be prime, semi-prime, o r right Noether-ian. We also determine the nil radical B (A, γ) of I(A, γ) for a large class of rings A, and show that under certain finiteness conditions, IiA, γ)/B (A, γ) i s isomorphic to the direct product of the incidence algebras of the strongly connected components of γ