使用图拉普拉斯矩阵的第二特征向量对稀疏图进行划分
Partitioning Sparse Graphs using the Second Eigenvector of their Graph Laplacian
AI总结：
本文利用图拉普拉斯矩阵的第二特征向量将图划分为两个大连通部分和一个小的分隔符集合，并扩展为层次细分，通过Lanczos算法高效处理稀疏图，实验验证了其有效性。
摘要：
将图划分为三部分，其中两部分规模较大且连通，第三部分为较小的“分隔符”集合，这有助于提升许多组合算法的性能。本文通过仅由关联矩阵定义的矩阵（称为图拉普拉斯矩阵）的第二特征向量来实现这一划分。对于稀疏图，该特征向量可使用Lanczos算法高效计算。此图划分算法被扩展以提供图的完整层次细分。该方法已实现，并针对简单测试问题和若干网格图获得了数值结果。
英文摘要：
Partitioning a graph into three pieces, with two of them large and connected, and the third a small ``separator'' set, is useful for improving the performance of a number of combinatorial algorithms. This is done using the second eigenvector of a matrix defined solely in terms of the incidence matrix, called the graph Laplacian. For sparse graphs, the eigenvector can be efficiently computed using the Lanczos algorithm. This graph partitioning algorithm is extended to provide a complete hierarchical subdivision of the graph. The method has been implemented and numerical results obtained both for simple test problems and for several grid graphs.
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。
