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The term signed graph is applied occasionally to graphs in which each edge has a weight, w(e) = +1 or −1. These are not the same kind of signed graph; they are weighted graphs with a restricted weight set. The difference is that weights are added, not multiplied. The problems and methods are completely different.
The 1980 monograph Spectra of Graphs [16] by Cvetković, Doob, and Sachs summarised nearly all research to date in the area. In 1988 it was updated by the survey Recent Results in the Theory of Graph Spectra. [17] The 3rd edition of Spectra of Graphs (1995) contains a summary of the further recent contributions to the subject. [15]
Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc.
The layout uses the eigenvectors of a matrix, such as the Laplace matrix of the graph, as Cartesian coordinates of the graph's vertices. The idea of the layout is to compute the two largest (or smallest) eigenvalues and corresponding eigenvectors of the Laplacian matrix of the graph and then use those for actually placing the nodes.
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter [ 1 ] and is one of the early examples of modern ...
Waterfall plots are often used to show how two-dimensional phenomena change over time. [1] A three-dimensional spectral waterfall plot is a plot in which multiple curves of data, typically spectra, are displayed simultaneously. Typically the curves are staggered both across the screen and vertically, with "nearer" curves masking the ones behind.
Switching {X,Y} in a graph. A two-graph is equivalent to a switching class of graphs and also to a (signed) switching class of signed complete graphs.. Switching a set of vertices in a (simple) graph means reversing the adjacencies of each pair of vertices, one in the set and the other not in the set: thus the edge set is changed so that an adjacent pair becomes nonadjacent and a nonadjacent ...
Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry ...