The electrical resistance of a graph captures its commute and cover times
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Note: Random walks and the effective resistance sum rules
Discrete Applied Mathematics
A class of graph-geodetic distances generalizing the shortest-path and the resistance distances
Discrete Applied Mathematics
Information Sciences: an International Journal
The asymptotic behavior of some indices of iterated line graphs of regular graphs
Discrete Applied Mathematics
A recursion formula for resistance distances and its applications
Discrete Applied Mathematics
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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.