  This investigation covers the effects of variable exchange interactions on
the spin dynamics of the zig-zag honeycomb lattice. Using a Holstein-Primakoff
expansion of the Heisenberg Hamiltonian with easy-axis anisotropy, we
characterize the effects of multiple nearest-neighbor and next-nearest-neighbor
interactions with asymmetry within the context of a frustrated and
non-frustrated zig-zag magnetic configuration. Furthermore, we compare to the
known inelastic neutron scattering data for the proximate quantum spin liquid
$\alpha$-RuCl$_3$, and we provide insight into the evolution of the spin
dynamics, showing that the Heisenberg interaction dominates the majority of the
spin excitation behavior. By analyzing the frustrated system with multiple
interactions, direction-dependent Dirac nodes present themselves, and we can
demonstrate that a standard Heisenberg model can accurately describe the
observed magnon spectra.
