| The sandpile group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. In this paper, we determined the structure of the sandpile group on the 4×n twisted bracelets K4,n[(12)] , K4,n[(123)] and m×n twisted bracelets Km,n[(12)] . The main results are(1) The sandpile group of bracelets graph K4,n[(12)] and K4,n[(123)] :(1a). The sandpile group of K4,n[(12)] is isomorphic to, for odd integer n=(?). For even integer n = 2m, it is isomorphic to (?), sm = (?).(1b). The sandpile group of K4,n[(123)] is isomorphic to, for odd integer n = 2m + 1, (?). For even integer n = 2m, itis isomorphic to (?).This part of paper has been published in " Journal of Natural Science of Hunan Normal University". The structure of the sandpile group on the 4×n twisted bracelets are determined by K4,n[(1)], K4,n[(12)], K4,n[(123)], K4,n[(1234)], K4,n[(12)(34)]. The sandpile group of K4,n[(1)]≌K4×Cn is isomorphic to the direct sum of three to five cyclic groups. So the sandpile group on the 4×n twisted bracelets are isomorphic to the direct sum of three to five cyclic groups. (2) The sandpile group of bracelets graph Km,n[(12)]:(2a). For odd integer n = 2t + 1, it is isornorphic to (?)(2b). For even integer n = 2t, it is isomorphic to (?)...
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