Over the past few decades, conventional first-order topological insulators (TIs) have attracted much attention in condensed matter physics. In contrast to conventional TIs with gapless edge states, two-dimensional higher-order topological insulators (2D HOTIs) host zero-dimensional in-gap corner states, which are spatially localized at geometric corners and energetically separated from both edge and bulk bands. Due to their localized nature and weak coupling to propagating states in external leads, these corner states do not naturally form robust transport channels, making their detection via direct current measurements challenging. Optical conductivity provides valuable information on charge dynamics and intrinsic band structures, which is often difficult to access through transport measurements. It is therefore natural to ask whether optical transitions among isolated corner states, edge states, and bulk states can generate distinct optical conductivity signatures.

In this work, we take finite-size hexagonal nanodisks of the Kekulé lattice as a concrete example and identify the optical conductivity signatures of topological corner and edge states in 2D HOTIs. We numerically investigate the band structures and optical conductivity of hexagonal nanodisks with different side lengths. Specifically, for the side lengths L=4a_0+3na_0, where a_0 is the bond length and n=0,1,2,\ldots, the hexagonal nanodisks of the Kekulé lattice are confirmed to be 2D HOTIs. In this case, additional corner and edge states appear within the bulk band gap. Since optical transitions involving corner states, edge states, and bulk states provide additional channels within the bulk gap, they give rise to a characteristic low-frequency response in the optical conductivity. As a consequence, the optical conductivity exhibits additional resonant peaks in the low-frequency region, where it vanishes for trivial insulators. These additional resonant peaks are absent in trivial insulating nanodisks because there are no corresponding in-gap corner or edge states to participate in the optical transitions. Moreover, these distinctive features in the optical conductivity can be observed as long as the chemical potential is located in the bulk gap, and they are robust against disorder-induced scattering broadening. Our findings extend the theoretical understanding of the dynamical aspects of topological corner and edge states in 2D HOTIs. The additional resonant peaks in the optical conductivity provide a promising route toward the optical detection of corner and edge states in 2D HOTIs.