  We examine the combined effects of a Kekule coupling texture (KC) and a
Dzyaloshinskii-Moriya interaction (DMI) in a two-dimensional ferromagnetic
honeycomb lattice. By analyzing the gap closing conditions and the inversions
of the bulk bands, we identify the parameter range in which the system behaves
as a trivial or a nontrivial topological magnon insulator. We find four
topological phases in terms of the KC parameter and the DMI strength. We
present the bulk-edge correspondence for the magnons in a honeycomb lattice
with an armchair or a zigzag boundary. Furthermore, we find Tamm-like edge
states due to the intrinsic on-site interactions along the boundary sites. Our
results may have significant implications to magnon transport properties in the
2D magnets at low temperatures.
