  In a class of carbon-based materials called polymerized triptycene, which
consist of triptycene molecules and phenyls, exotic electronic structures such
as Dirac cones and flat bands arise from the kagome-type network. In this
paper, we theoretically investigate the tight-binding models for polymerized
triptycene, focusing on the origin of flat bands and the topological
properties. The mechanism of the existence of the flat bands is elucidated by
using the "molecular-orbital" representation, which we have developed in the
prior works. Further, we propose that the present material is a promising
candidate to realize the two-dimensional second-order topological insulator,
which is characterized by the boundary states localized at the corners of the
sample. To be concrete, we propose two methods to realize the second-order
topological insulator, and elucidate the topological properties of the
corresponding models by calculating the corner states as well as the bulk
topological invariant, namely the $\mathbb{Z}_3$ Berry phase.
