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In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the notion of balance appeared first in a mathematical paper of Frank Harary in 1953. [1]
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]
The upper half of this diagram shows the frequency spectrum of a modern switching power supply which employs spread spectrum. The lower half is a waterfall plot showing the variation of the frequency spectrum over time during the power supply's heating up period. Spectrogram and 3 styles of waterfall plot of a whistled sequence of 3 notes vs time
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 cycle of length 5 is an srg(5, 2, 0, 1). The Petersen graph is an srg(10, 3, 0, 1). The Clebsch graph is an srg(16, 5, 0, 2). The Shrikhande graph is an srg(16, 6, 2, 2) which is not a distance-transitive graph. The n × n square rook's graph, i.e., the line graph of a balanced complete bipartite graph K n,n, is an srg(n 2, 2n − 2, n −
An example connected graph, with 6 vertices. Partitioning into two connected graphs. In multivariate statistics, spectral clustering techniques make use of the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions. The similarity matrix is provided as an input and ...
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A planar graph cannot contain K 3,3 as a minor; an outerplanar graph cannot contain K 3,2 as a minor (These are not sufficient conditions for planarity and outerplanarity, but necessary). Conversely, every nonplanar graph contains either K 3,3 or the complete graph K 5 as a minor; this is Wagner's theorem. [9] Every complete bipartite graph.