关于Rossi, Frasca和Fagnani论文《toroidal graphs的平均电阻》的注记
Note on "Average resistance of toroidal graphs" by Rossi, Frasca and Fagnani
AI总结：
本文基于之前研究，探讨了toroidal网格的平均电阻与图拓扑和维度的关系，并指出其与随机游走的平均击中时间相关联。
摘要：
在我们最近的论文W.S. Rossi, P. Frasca和F. Fagnani的《toroidal graphs的平均电阻》中，我们研究了d维toroidal网格的平均电阻如何依赖于图拓扑和图的维度。我们的结果基于电阻与拉普拉斯特征值之间的联系。在本文中，我们将我们的工作置于图上随机游走的文献背景中。确实，d维toroidal网格的平均有效电阻与该网格上简单随机游走的平均击中时间成正比。如果d≥3，则平均电阻可以被统一地限制在节点数量中，并且对于大的d，其值的顺序为1/d。
英文摘要：
In our recent paper W.S. Rossi, P. Frasca and F. Fagnani, "Average resistance of toroidal graphs", SIAM Journal on Control and Optimization, 53(4):2541--2557, 2015, we studied how the average resistances of $d$-dimensional toroidal grids depend on the graph topology and on the dimension of the graph. Our results were based on the connection between resistance and Laplacian eigenvalues. In this note, we contextualize our work in the body of literature about random walks on graphs. Indeed, the average effective resistance of the $d$-dimensional toroidal grid is proportional to the mean hitting time of the simple random walk on that grid. If $d\geq3 $, then the average resistance can be bounded uniformly in the number of nodes and its value is of order $1/d$ for large $d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
