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Spectral layout drawing of random small-world network. For comparison, the same graph plotted as spring graph drawing. Spectral layout is a class of algorithm for drawing graphs. The layout uses the eigenvectors of a matrix, such as the Laplace matrix of the graph, as Cartesian coordinates of the graph's vertices.
In 1910 Hans Oswald Rosenberg published a diagram plotting the apparent magnitude of stars in the Pleiades cluster against the strengths of the calcium K line and two hydrogen Balmer lines. [3] These spectral lines serve as a proxy for the temperature of the star, an early form of spectral classification.
The smallest pair of cospectral mates is {K 1,4, C 4 ∪ K 1}, comprising the 5-vertex star and the graph union of the 4-vertex cycle and the single-vertex graph [1]. The first example of cospectral graphs was reported by Collatz and Sinogowitz [2] in 1957. The smallest pair of polyhedral cospectral mates are enneahedra with eight vertices each ...
The Spectral layout is based on the spectral properties of the graph's adjacency matrix. It uses the eigenvalues and eigenvectors of the adjacency matrix to position nodes in a low-dimensional space. Spectral layout tends to emphasize the global structure of the graph, making it useful for identifying clusters and communities. [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.
In Fluorescence lifetime and spectral imaging, phasor can be used to visualize the spectra and decay curves. [1] [2] In this method the Fourier transformation of the spectrum or decay curve is calculated and the resulted complex number is plotted on a 2D plot where the X-axis represents the real component and the Y-axis represents the imaginary ...
where ∂ out (S) is the outer boundary of S, i.e., the set of vertices in V(G) \ S with at least one neighbor in S. [3] In a variant of this definition (called unique neighbor expansion) ∂ out (S) is replaced by the set of vertices in V with exactly one neighbor in S. [4] The vertex isoperimetric number h in (G) of a graph G is defined as
Described by the amount, wavelength interval, and width of spectral bands in which the sensor conducts wavelength measurements, a sensor with high spectral resolution would mean that it is able to capture a spectrum of light and divides it into hundreds or thousands of narrow spectral bands or channels with typical widths up to 10 and 20 nm. [11]