A graph theoretic model for the derived categories of gentle algebras and their homological bilinear forms

We formulate a simple model for the bounded derived category of gentle algebras in terms of marked ribbon graphs and their walks, in order to analyze indecomposable objects, Auslander-Reiten triangles and homological bilinear forms, and to provide some relevant derived invariants in a graph theoretic setting. Among others, we exhibit the non-negativity and Dynkin type of the homological quadratic form of a gentle algebra, describe its roots as the classes of indecomposable perfect complexes in the Grothendieck group, and express its Coxeter polynomial in terms of the Avella-Alaminos Geiss invariant. We also derive some consequences for Brauer graph algebras.