Resistance distance and the normalized Laplacian spectrum

By Haiyan Chen, Fuji Zhang. Discrete Applied Mathematics, 2007

344 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/f0975f573c625a298e7acc860eb53e9af53acb7c

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Citations in the corpus

Unicyclic graphs with extremal Kirchhoff index (2008, methodology): “It is computed in various ways: in algebra using generalized inverse of Laplacian matrix [2], Laplacian eigenvalues and eigenvectors [13] and normalized Laplacian eigenvalues and eigenvectors [16]; in combinatorics using spanning trees and spanning bi-trees [14]; in probability using random walks [1”
ON THE NORMALISED LAPLACIAN SPECTRUM, DEGREE-KIRCHHOFF INDEX AND SPANNING TREES OF GRAPHS (2015, unknown): “T heorem 2.3 [6].”
On resistance-distance and Kirchhoff index (2009, background): “Recently, Chen and Zhang [ 16 ] introduced a quantity Kf ∗ (G), which is defined as:”
Distribution of shortest path lengths in subcritical Erdős-Rényi networks. (2018, background): “It was shown that the resistance distance between nodes i and j in a network can be decomposed in terms of the eigenvalues and eigenvectors of the normalized Laplacian matrix of the network [52,53].”
Algebraic aspects of the normalized Laplacian (2016, background): “For more information on this parameter see Chen and Zhang [11].”
Bounds for the Kirchhoff index via majorization techniques (2013, methodology): “The degree-Kirchhoff index was proposed by Chen and Zhang in [5], defined as”