  In the first part we study nearly Frobenius algebras. The concept of nearly
Frobenius algebras is a generalization of the concept of Frobenius algebras.
Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove
that the known constructions: direct sums, tensor, quotient of nearly Frobenius
algebras admit natural nearly Frobenius structures.
  In the second part we study algebras associated to some families of quivers
and the nearly Frobenius structures that they admit. As a main theorem, we
prove that an indecomposable algebra associated to a bound quiver $(Q,I)$ with
no monomial relations admits a non trivial nearly Frobenius structure if and
only if the quiver is $\overrightarrow{\mb{A}_n}$ and I=0. We also present an
algorithm that determines the number of independent nearly Frobenius structures
for Gentle algebras without oriented cycles.
