In this work, we study the propagation of sound waves in a honeycomb waveguide network loaded with Helmholtz resonators (HRs). By using a plane wave approximation in each waveguide we obtain a first-principle modeling of the network, which is an exact mapping to the graphene tight-binding Hamiltonian. We show that additional Dirac points appear in the band diagram when HRs are introduced at the network nodes. It allows to break the inversion (sub-lattice) symmetry by tuning the resonators, leading to the appearence of edge modes that reflect the configuration of the zigzag boundaries. Besides, the dimerization of the resonators also permits the formation of interface modes located in the band gap, and these modes are found to be robust against symmetry preserving defects. Our results and the proposed networks reveal the additional degree of freedom bestowed by the local resonance in tuning the properties of not only acoustical graphene-like structures but also of more complex systems.