Resistance distance and the normalized Laplacian spectrum
Resistance distance and the normalized Laplacian spectrum
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DOI：
10.1016/j.dam.2006.09.008
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发表时间：
2007-03-15
影响因子：
1.1
通讯作者：
Zhang, Fuji
中科院分区：
文献类型：
--
作者：
Chen, Haiyan;Zhang, Fuji
众所周知，电网络中任意两个顶点之间的电阻距离可以用与该网络相关的组合拉普拉斯矩阵的特征值和特征向量来表示。通过对这个矩阵的研究，人们证明了电阻距离的许多性质。但近年来，另一种与谱几何和随机游走中的矩阵相一致的称为归一化拉普拉斯矩阵的矩阵引起了人们的关注[Chung， F.R.K, spectral Graph Theory, American Mathematical Society: Providence， RI, 1997]。因为许多人认为基于这个矩阵的量可以更真实地反映图的结构和性质。本文不仅证明了电阻距离可以自然地用G的归一化拉普拉斯特征值和特征向量表示，而且引入了一个与归一化拉普拉斯谱密切相关的新指标。最后，我们发现了著名的基尔霍夫指数与新指数之间的非平凡关系。(c) 2006 Elsevier B.V.版权所有
It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix associated with the network. By Studying this matrix, people have proved many properties of resistance distances. But in recent years, the other kind of matrix, named the normalized Laplacian, which is consistent with the matrix in spectral geometry and random walks [Chung, F.R.K., Spectral Graph Theory, American Mathematical Society: Providence, RI, 1997], has engendered people's attention. For many people think the quantities based on this matrix may more faithfully reflect the structure and properties of a graph. In this paper, we not only show the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also introduce a new index which is closely related to the spectrum of the normalized Laplacian. Finally we find a non-trivial relation between the well-known Kirchhoff index and the new index. (c) 2006 Elsevier B.V. All rights reserved.