Electron-phonon coupling in Kekul\'{e}-ordered graphene

Breaking the intrinsic chirality of quasiparticles in graphene enables the emergence of new and intriguing phases. One such paradigmatic example is the bond density wave, which leads to a Kekul\'{e}-ordered structure and underpins exotic electronic states where electron-phonon interactions can play a fundamental role. Here, it is shown that the relevant physics of these correlations can be resolved locally, according to the behavior of interatomic characteristics. For this purpose a robust distance-dependent framework for describing electronic structure of graphene with Kekul\'{e} bond order is presented. Given this insight, the strength of electron-phonon interactions is found to scale linearly with the electronic coupling, contributing to a uniform picture of this relationship in distorted graphene structures. Moreover, it is shown that the introduced distortion yields a strongly non-uniform spatial distribution of the pairing strength that eventually leads to the induction of periodically distributed domains of enhanced electron-phonon coupling. These findings help elucidate certain peculiar aspects of phonon-mediated phenomena in graphene, particularly the associated superconducting phase, and offer potential pathways for their further engineering.