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The study of topological states in electronic structures, which allows robust transport properties against impurities and defects, has been recently extended to the realm of elasticity. This work shows that nontrivial topological flexural edge states located on the free boundary of the elastic graphene-like metamaterial can be realized without breaking the time reversal, mirror, or inversion symmetry of the system. Numerical calculations and experimental studies demonstrate the robust transport of flexural waves along the boundaries of the designed structure. The topological edge states on the free boundary are not limited by the size of the finite structure, which can reduce the scale of the topological state system. In addition, unlike the edge states localized on the free boundary in graphene where the group velocity is zero, the edge states on the elastic metamaterial plate have propagation states with non-zero group velocity. There is a frequency range for the edge states, and we introduce the concept of Shannon entropy for elastic waves and use it to assess the frequency range of the edge states in graphene-like elastic metamaterials. This work represents a relevant advance in the study of elastic wave topological states, providing a theoretical basis for engineering applications such as vibration reduction and vibration isolation of mechanical structures., Comment: 22 pages, 8 figures, and 12 pages of supplementary information