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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 Seidel matrix of G is also the adjacency matrix of a signed complete graph K G in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and K G. The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs.
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]
They considered an operation called a 2-lift that takes a -regular graph with vertices and a sign on each edge, and produces a new -regular graph ′ on vertices. Bilu & Linial conjectured that there always exists a signing so that every new eigenvalue of G ′ {\displaystyle G'} has magnitude at most 2 d − 1 {\displaystyle 2{\sqrt {d-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 − 2, 2). The parameters for n = 4 coincide with those of the Shrikhande graph, but the two ...
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.
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 ...