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A signed graph is the special kind of gain graph in which the gain group has order 2. The pair (G, B(Σ)) determined by a signed graph Σ is a special kind of biased graph. The sign group has the special property, not shared by larger gain groups, that the edge signs are determined up to switching by the set B(Σ) of balanced cycles. [19]
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]
A common format is a graph with two geometric dimensions: one axis represents time, and the other axis represents frequency; a third dimension indicating the amplitude of a particular frequency at a particular time is represented by the intensity or color of each point in the image.
Spectral graph theory relates properties of a graph to a spectrum, i.e., eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix. Imbalanced weights may undesirably affect the matrix spectrum, leading to the need of normalization — a column/row scaling of the matrix entries ...
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 ...
The separated parts of Hafner's construction. In mathematical graph theory, the Higman–Sims graph is a 22-regular undirected graph with 100 vertices and 1100 edges. It is the unique strongly regular graph srg(100,22,0,6), where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors. [2]
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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 −