On $Q$-deformations of Postnikov-Shapiro algebras

For any given loopless graph $G$, we introduce $Q$ - deformations of its Postnikov-Shapiro algebras counting spanning trees, counting spanning forests and $Q$ - deformations of internal algebra of $G$. We determine the total dimension of the algebras; our proof also gives a new proof of the formula for the total dimensions of the usual Postnikov-Shapiro algebras. Furthermore, we construct “square-free” definition of usual internal algebra of $G$.