  Given any simple biorientable graph it is shown that there exists a weak
{*}-Hopf algebra constructed on the vector space of graded endomorphisms of
essential paths on the graph. This construction is based on a direct sum
decomposition of the space of paths into orthogonal subspaces one of which is
the space of essential paths. Two simple examples are worked out with certain
detail, the ADE graph $A_{3}$ and the affine graph $A_{[2]}$. For the first
example the weak {*}-Hopf algebra coincides with the so called double triangle
algebra. No use is made of Ocneanu's cell calculus.
