Resistance distance and the normalized Laplacian spectrum

Abstract

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.

Hitting Times, Cover Cost, and the Wiener Index of a Tree

“…It was observed in [CZ07] that the latter expression 2m λ =1 1 1−λ equals K π 2 (G), but apparently the resulting fact that CC π (x) = K π 2 (G) has not been noticed before. It is thus worth pointing out this triple equality:…”

Lower bounds of the Kirchhoff and degree Kirchhoff indices

“…, n − 1 and (1), we obtain that A = 2m n−2 . By substituting A = 2m n−2 in (6) we arrive at (5). Having in mind that equality in (3), for n := n − 1, occurs if and only if a 1 = a 2 = · · · = a n−1 , equality in (5) …”

Resistance Distance, Information Centrality, Node Vulnerability and Vibrations in Complex Networks

“…Then, the average path lengthl is given byl = 2W(G)/n(n− 1). The analogue of the Wiener index in the context of the resistance distance matrix is known as the Kirchhoff index Kf and is defined as Kf = i<j Ω ij [6,25,35,36,38]. It is known that Kf can be expressed in terms of the Laplacian eigenvalues as follows [35]:…”