Search results
Results from the WOW.Com Content Network
Spectral graph theory emerged in the 1950s and 1960s. Besides graph theoretic research on the relationship between structural and spectral properties of graphs, another major source was research in quantum chemistry , but the connections between these two lines of work were not discovered until much later. [ 15 ]
Fan-Rong King Chung Graham (Chinese: 金芳蓉; pinyin: Jīn Fāngróng; born October 9, 1949), known professionally as Fan Chung, is a Taiwanese-born American mathematician who works mainly in the areas of spectral graph theory, extremal graph theory and random graphs, in particular in generalizing the Erdős–Rényi model for graphs with general degree distribution (including power-law ...
The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is important in spectral graph theory. It is widely applied in the recent study of graph structured learning algorithms, such as the widely employed convolutional networks.
In mathematics, spectral theory deals with attempts to understand operators, graphs and dynamical systems by means of the spectrum of eigenvalues associated with the system. The classical examples of spectra are the vibration modes of a violin string or the spectrum of a hydrogen atom .
This quantity is studied in the context of spectral graph theory. More precisely, let G be a graph with n vertices. It is assumed that G is a simple graph, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let , =, …,, be the eigenvalues of A. Then the energy of the graph is defined as:
For every graph G, there exists a highly irregular graph H containing G as an induced subgraph. [ 3 ] This last observation can be considered analogous to a result of Dénes Kőnig , which states that if H is a graph with greatest degree r , then there is a graph G which is r -regular and contains H as an induced subgraph.
Fan Chung has developed an extensive theory using a rescaled version of the Laplacian, eliminating the dependence on the number of vertices, so that the bounds are somewhat different. [ 7 ] In models of synchronization on networks, such as the Kuramoto model , the Laplacian matrix arises naturally, so the algebraic connectivity gives an ...