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In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. [1] It is a result of studies of linear algebra and the solutions of systems of linear equations and their ...
The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional ...
In mathematics, the spectrum of a matrix is the set of its eigenvalues. [ 1 ] [ 2 ] [ 3 ] More generally, if T : V → V {\displaystyle T\colon V\to V} is a linear operator on any finite-dimensional vector space , its spectrum is the set of scalars λ {\displaystyle \lambda } such that T − λ I {\displaystyle T-\lambda I} is not invertible .
The smoothed periodogram is sometimes referred to as a spectral plot. [11] [12] Periodogram-based techniques introduce small biases that are unacceptable in some applications. Other techniques that do not rely on periodograms are presented in the spectral density estimation article.
Since = if and only if the graph is bipartite, we will refer to the graphs that satisfy this alternative definition but not the first definition bipartite Ramanujan graphs. If G {\displaystyle G} is a Ramanujan graph, then G × K 2 {\displaystyle G\times K_{2}} is a bipartite Ramanujan graph, so the existence of Ramanujan graphs is stronger.
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix. This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number C such that the degree of every vertex of the graph is smaller than C). In this case, for the graph G define:
Spectral analysis or spectrum analysis is analysis in terms of a spectrum of frequencies or related quantities such as energies, eigenvalues, etc. In specific areas it may refer to: Spectroscopy in chemistry and physics, a method of analyzing the properties of matter from their electromagnetic interactions