  In this article we further investigate the construction of graph coherent
states, first introduced in [1], in the context of loop quantum gravity. We
specifically investigate the possibility of defining a family of graph coherent
states adapted to the canonical loop quantum gravity Hamiltonian. After
discussing various aspects of the general framework and the choice of
operators, we use the Euclidean part of the Hamiltonian operator to propose a
generator of the generalized canonical structure, necessary to define the
coherent states. We then apply the construction procedure, leading to a new
family of graph coherent states partially adapted to the gravity Hamiltonian in
loop quantum gravity.
