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