  In this paper we determine the first Hochschild homology and cohomology with
different coefficients for gentle algebras and we give a geometrical
interpretation of these (co)homologies using the ribbon graph of a gentle
algebra as defined in earlier work by the second author. We give an explicit
description of the Lie algebra structure of the first Hochschild cohomology of
gentle and Brauer graph algebras (with multiplicity one) based on trivial
extensions of gentle algebras and we show how the Hochschild cohomology is
encoded in the Brauer graph. In particular, we show that except in one
low-dimensional case, the resulting Lie algebras are all solvable.
