Since the discovery of the long-range ferromagnetic order in two-dimensional and multi-layered van der Waals crystals, and the observation of a nontrivial topology of the magnon bulk bands in the chromium trihalides, the bosonic honeycomb lattices have drawn significant attention within the condensed matter community. In this thesis, we employ a Heisenberg model with a Dzyaloshinsky- Moriya interaction in a honeycomb ferromagnetic lattice to study the properties of bulk and edge spin-wave excitations (magnon). By the Holstein- Primakoff transformations in the linear spin-wave approximation, the spin Hamiltonian is written as the bosonic equivalent of the Haldane model for spinless fermions. We present a simple bosonic tight binding formalism which allows us to obtain analytical solutions for the energy spectrum and wavefunctions. We investigate three basic boundaries in the honeycomb lattice: zigzag, bearded and armchair, and we derive analytical expressions for the energy band structure and wavefunctions for the bulk and edge states, and with both zero and nonzero Dzyaloshinsky- Moriya interaction. We find that in a lattice with a boundary, the intrinsic on-site interactions along the boundary sites generate an effective defect and this gives rise to Tamm-like edge states. If a nontrivial gap is induced, both Tamm-like and topologically protected edge states appear in the band structure. The effective defect can be strengthened by an external on-site potential, and the dispersion relation, velocity and magnon density of the edge states all become tunable. We also investigate the bond modulation in the bosonic Haldane model, where by introducing a Kekule bond modulation and with the analysis of the gap closing conditions and the bulk band inversions, we find a rich topological phase diagram for this system yet to be discovered. We identify four topological phases, verified by a numerical calculation of the Chern number, in terms of the Kekule modulation parameter and the Dzyaloshinsky- Moriya interaction. We present the bulk-edge correspondence for the magnons in a honeycomb lattice for both armchair and zigzag boundaries. We believed that our study in this thesis will be important for possible applications of magnons in data process devices such as magnonics.