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This is the second paper in the series to introduce a graphical method toloop quantum gravity. We employ the graphical method as a powerful tool tocalculate the actions of the Euclidean Hamiltonian constraint operator and theso-called inverse volume operator on spin network states with trivalentvertices. Both of the operators involve the co-triad operator which containsholonomies by construction. The non-ambiguous, concise and visual characters ofour graphical method ensure the rigour for our calculations. Our resultsindicate some corrections to the existing results in literatures.

Graphical method in loop quantum gravity: II. The Hamiltonian constraint and inverse volume operators