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MAT 280 Harmonic Analysis on Graphs & Networks Reference Page (Fall 2019)
The general introductory references
For general introduction to graphs and networks and significant applications:
D. Cvetković, P. Rowlinson, and S. Simić: An Introduction to the Theory of Graph Spectra, Vol. 75, London Mathematical Society Student Texts, Cambridge Univ. Press, 2010.
On the sum of even powers of reciprocal eigenvalues of the Laplacian (sometimes
called as the Rayleigh function), please refer to the following paper of main
and the references therein:
The fast algorithms for the Laplacian eigenvalues and eigenfunctions
via commuting integral operators and the Fast Multipole Methods, see
Allen Xue's Ph.D. dissertation.
Of course, it is indispensable to check the following ground breaking papers:
Lecture 3: Basics of Graph Theory: Graph Laplacians
This lecture is based on the following material:
J. H. van Lint and R. M. Wilson: A Course in Combinatorics, 2nd Ed.,
Chap. 31, Cambridge Univ. Press, 2001.
H. Urakawa: "Spectral geometry and graph theory," Ouyou Suuri (Bulletin of the Japan Society of Industrial and Applied Mathematics), vol. 12, no. 1, pp.29-45, 2002.
F. R. K. Chung: Spectral Graph Theory, Sec. 1.1-1.3, AMS, 1997.
I explained that the graph Laplacian of a simple path graph is the same as the matrix used in the Discrete Cosine Transform Type II (DCT-II).
More precisely, the eigenvectors of that graph Laplacian are the DCT-II
basis vectors. Subtle changes in the discrete boundary condition generates
matrices whose eigenvectors are DCT-I, III, IV, ... More about this interesting
relationship between the eigenvectors of discrete boundary value problems
and the various types of DCT and Discrete Sine Transform (DST) can be
found in
Books by Cvetković et al., Chung, Bapat, Brouwer and W. H. Haemers
contain relevant chapters on graph Laplacian eigenvalues.
But there are many good survey papers on graph Laplacian eigenvalues. Here are
some representative ones.
The books on human vision I referred to during my lecture are:
D. Hubel: Eye, Brain, and Vision, Scientific American Library, 1995.
D. Marr: Vision: A Computational Investigation into the Human Representation and Processing of Visual Information, W. H. Freeman & Co., 1982, republished by The MIT Press, 2010.
Finally, a nice description between the Laplacian eigenpairs and
a simple 1D wave equation that explains why the Laplacian eigenvalues (e.g.,
sound frequencies) reflect geometry of the domain (e.g., the length of a
guitar string), see "Section 2: History of Laplacian Eigenvalue Problems -
Spectral Geometry" of the following tutorial of mine:
For our analysis on dendritic trees including the phase transition phenomena and morphological feature extraction from dendritic trees using their graph
Laplacians, see:
For the excellent read on the random walks on graphs and networks, see:
P. G. Doyle and J. L. Snell: Random Walks and
Electric Networks, The Carus Mathematical Monographs, no. 22,
Math. Assoc. Amer., 1984. Also available via arXiv:math/0001057v1.
Raphy Coifman's group at Yale developed the concept of diffusion geometry
and applied for hundreds of practical problems. Some of their early effort was
summarized in the following special issue of the journal, which include
the above article on the diffusion maps:
Earth Mover's Distance and its more general version
Wasserstein distances have become very popular in many applications. There are too many references on these distances. Below are just a few picks:
There are too many wavelet-like transform constructions on graphs and networks to list them all here. The following are good surveys of some of those efforts:
I also mentioned the concept of the Minimum Description Length (MDL) criterion for model selection, which was originally proposed by Jorma Rissanen, and is quite interesting and important in many statistical inference problems. I would suggest the following paper and books to know the MDL principle better: