Dual-band valley-protected topological edge states in graphene-like phononic crystals with waveguide Qianlong Kang, Fujia Chen, Hongyong Mao, Keya Zhou, Kai Guo, Shutian Liu and Zhongyi Guo Eur. Phys. J. B, 96 3 (2023) 40 Published online: 31 March 2023 DOI: 10.1140/epjb/s10051-023-00503-4