Topic
Spectral graph theory
About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.
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TL;DR: In this paper, a quantum walk defined by digraphs (mixed graphs) is proposed, where the discriminant of the quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices.
Abstract: We propose a quantum walk defined by digraphs (mixed graphs). This is like Grover walk that is perturbed by a certain complex-valued function defined by digraphs. The discriminant of this quantum walk is a matrix that is a certain normalization of generalized Hermitian adjacency matrices. Furthermore, we give definitions of the positive and negative supports of the transfer matrix and exhibit explicit formulas of supports of their square. Also, we provide tables on the identification of digraphs by their eigenvalues.
12 citations
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31 Mar 2016TL;DR: This paper studies the concept of spectrum folding (aliasing) for graph signals under the downsample-then-upsample operation with a special eigenvector structure that is unique to the adjacency matrix of M-block cyclic matrices.
Abstract: Signal processing on graphs finds applications in many areas. Motivated by recent developments, this paper studies the concept of spectrum folding (aliasing) for graph signals under the downsample-then-upsample operation. In this development, we use a special eigenvector structure that is unique to the adjacency matrix of M-block cyclic matrices. We then introduce M-channel maximally decimated filter banks. Manipulating the characteristics of the aliasing effect, we construct polynomial filter banks with perfect reconstruction property. Later we describe how we can remove the eigenvector condition by using a generalized decimator. In this study graphs are assumed to be general with a possibly non-symmetric and complex adjacency matrix.
12 citations
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TL;DR: Wang et al. as discussed by the authors presented a novel method that performs the localized graph convolutional filtering on hyperspectral image (HSI) classification based on spectral graph theory.
Abstract: The nascent graph representation learning has shown superiority for resolving graph data. Compared to conventional convolutional neural networks, graph-based deep learning has the advantages of illustrating class boundaries and modeling feature relationships. Faced with hyperspectral image (HSI) classification, the priority problem might be how to convert hyperspectral data into irregular domains from regular grids. In this regard, we present a novel method that performs the localized graph convolutional filtering on HSIs based on spectral graph theory. First, we conducted principal component analysis (PCA) preprocessing to create localized hyperspectral data cubes with unsupervised feature reduction. These feature cubes combined with localized adjacent matrices were fed into the popular graph convolution network in a standard supervised learning paradigm. Finally, we succeeded in analyzing diversified land covers by considering local graph structure with graph convolutional filtering. Experiments on real hyperspectral datasets demonstrated that the presented method offers promising classification performance compared with other popular competitors.
12 citations
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TL;DR: In this paper, the Godsil-McKay algorithm was ported to signed graphs, and it was shown that with suitable adaption, such algorithms can be successfully ported to cospectral switching nonisomorphic signed graphs.
Abstract: A well--known fact in Spectral Graph Theory is the existence of pairs of isospectral nonisomorphic graphs (known as PINGS). The work of A.J. Schwenk (in 1973) and of C. Godsil and B. McKay (in 1982) shed some light on the explanation of the presence of isospectral graphs, and they gave routines to construct PINGS. Here, we consider the Godsil-McKay--type routines developed for graphs, whose adjacency matrices are $(0,1)$-matrices, to the level of signed graphs, whose adjacency matrices allow the presence of $-1$'s. We show that, with suitable adaption, such routines can be successfully ported to signed graphs, and we can build pairs of cospectral switching nonisomorphic signed graphs.
12 citations
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TL;DR: A lower bound for the energy of a symmetric matrix partitioned into blocks is introduced and some computational experiments are presented in order to show that the obtained lower bound is incomparable with the well known lower bound 2m.
12 citations