4_n手镯图的临界群.pdf

2009 3 Vo.l 32 No. 1 32 1 Journal of Natural Science ofH unan Norm al University M ar. , 2009 * 4 n 熊坤云, 侯耀平 ( , 4100 1) , . Lap lacian . 4 nK 4, n [ ( 12) ] K 4, n [ ( 123) ] , 3~ 5 . ; Laplacian; ; Sm ith O 1575 A 2009) O n the Sandp ile G roup of 4 n Tw isted B race lets X ION G K unyun, H OU Yaop ing ( College ofM athem atics and Com puter Science, H unan Norm alUn iversity, Changsha 4100 1, Ch ina) Ab stract The sandp ile group of a graph is anAbelian group that arises in several contexts, and a refinement of the number of spanning trees of the graph. It is subtle isomorphism invariant of the graph and closely connected w ith the graph Laplacian matrix. The abstract structures of the sandp ile groups on 4 n tw isted bracelets are deter m ined and it is show n that the sandp ile groups of those tw isted bracelets are awl ays isomorphic to the d irect sum of three to five cyclic groups. K ey w ord s sandp ile group; Laplacian matrix; the Sm ith normal form; bracelet graphs G = (V, E ) n. Lap lacianL (G) = D (G ) - A (G ), D (G ) = diag(d , 1 n n d , , d ) G, A ( G) G . L (G ) !