Let Δ be a finite quiver, that is, a finite directed graph, k be a commutative ring, and kΔ be the semigroup ring of paths in Δ. In this paper we compute the Hochschild homology of the following classes of algebras:

(1) kΔ/[mfr ]n, where [mfr ] is the arrow ideal;

(2) kΔ/I where I is an ideal generated by quadratic monomials.

In Section 2 we establish the notation, and recall a projective resolution of kΔ0, the degree 0 part in kΔ, over a monomial algebra which is due to Anick and Green. This resolution is then used in Section 3 as the building block in the construction of a resolution of a truncated or quadratic monomial algebra A, over its enveloping algebra (in the Hochschild sense), Ae. The fact that a monomial algebra possesses a fine grading then enables us to compute the homology.

The results obtained extend results by Liu and Zhang [3] and Geller, Reid and Weibel [2].