AI 中文总结
研究单纯复形拉普拉斯矩阵的史密斯标准型，通过建立邻接图沙堆群与该标准型的关系，计算单纯环面和\(k -\)树的相关标准型，经数值实验可视化其区分\(k -\)树的效果，并指出相关矩阵在拉普拉斯矩阵中的应用。
AI 中文摘要
受树的距离矩阵行列式公式推广到\(k -\)树的启发，我们研究了与某些单纯复形相关的拉普拉斯矩阵的史密斯标准型。我们发现邻接图的沙堆群与单纯复形拉普拉斯矩阵的史密斯标准型之间的关系。利用这些关系计算单纯环面和\(k -\)树的最高拉普拉斯矩阵的史密斯标准型。还提供数值实验来可视化这些代数不变量区分\(k -\)树的效果。最后指出，用于计算树的距离矩阵行列式的格雷厄姆 - 洛瓦兹 - 波拉克矩阵可用于树和块图的拉普拉斯矩阵情况。
英文摘要
Inspired by the generalization of the formula of determinant of the distance matrix of trees to $k$-trees, we study the Smith normal form of Laplacian matrices associated with some simplicial complexes. We find relations between sandpile groups of adjacency graphs and the Smith normal form of Laplacian matrices of simplicial complexes. We use such relations to calculate the Smith normal form of the highest Laplacian matrix of simplicial annuli and $k$-trees. %The low dimensional cases seems to be more difficult to calculate. We also provide numerical experiments to visualize how good are these algebraic invariants to distinguish $k$-trees. Finally, we point out that the Graham-Lovász-Pollak matrix, used to compute the determinant of the distance matrix of trees, can be used in the context of Laplacian matrices of trees and block graphs.
