On the first Hochschild cohomology group of an algebra

Let A = K Delta /I be a factor of a path algebra. We develop a strategy to compute dim H-1(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Delta, I). That allows us to connect dim H-1(A) with the rank and p-rank of the fundamental group pi (1)(Delta, I) of (Delta, I). We get explicit formulae for dim H-1(A), when every path in Delta parallel to an arrow belongs to I or when I is homogeneous.