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

Paths on graphs and associated quantum groupoids