更改

Deleting the row and column of \Delta corresponding with the sink yields the reduced graph Laplacian \Delta'. Then, when starting with a configuration z and toppling each vertex v a total of \mathbf{x}(v)\in\mathbb{N}_0 times yields the configuration z-\Delta'\boldsymbol{\cdot}~\mathbf{x}, where \boldsymbol{\cdot} is the contraction product. Furthermore, if \mathbf{x} corresponds to the number of times each vertex is toppled during the stabilization of a given configuration z, then

Deleting the row and column of \Delta corresponding with the sink yields the reduced graph Laplacian \Delta'. Then, when starting with a configuration z and toppling each vertex v a total of \mathbf{x}(v)\in\mathbb{N}_0 times yields the configuration z-\Delta'\boldsymbol{\cdot}~\mathbf{x}, where \boldsymbol{\cdot} is the contraction product. Furthermore, if \mathbf{x} corresponds to the number of times each vertex is toppled during the stabilization of a given configuration z, then

删除与汇对应的 Delta 行和列，得到简化图 Laplacian Delta’。然后，当以一个配置 z 开始并将每个顶点 v 在 mathbb { n } _ 0中的总和为 mathbf { x }(v)时，产生配置 z-Delta’粗体符号{ cdot } ~ mathbf { x } ，其中粗体符号{ cdot }是收缩积。此外，如果 mathbf { x }对应于在给定配置 z 的稳定过程中每个顶点被推翻的次数，则

删除与汇对应的 Delta 行和列，得到简化图 Laplacian Delta’。然后，当以一个配置 z 开始并将每个顶点 v 在 mathbb { n } _ 0中的总和为 \mathbf{x}(v)\in\mathbb{N}_0 时，产生配置 z-\Delta'\boldsymbol{\cdot}~\mathbf{x}，其中\boldsymbol{\cdot}是收缩积。此外，如果 \mathbf{x} 对应于在给定配置 z 的稳定过程中每个顶点被推翻的次数，则

To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math> without loops, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with  

To generalize the sandpile model from the rectangular grid of the standard square lattice to an arbitrary undirected finite multigraph <math>G=(V,E)</math> without loops, a special vertex <math>s\in V</math> called the ''sink'' is specified that is not allowed to topple. A ''configuration'' (state) of the model is then a function <math>z:V\setminus\{s\}\rightarrow\mathbb{N}_0</math> counting the non-negative number of grains on each non-sink vertex. A non-sink vertex <math>v\in V\setminus\{s\}</math> with