On the Laplacian spectra of product graphs

Barik, Sasmita; Bapat, R.B.; Pati, S.

doi:10.2298/aadm150218006b

Cited by 43 publications

(29 citation statements)

References 24 publications

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“…(We will come back to this point later.) In the meantime, however, there are partial regularities known for Laplacian spectra of direct product graphs that resemble those of Cartesian product graphs, especially when either factor graph is regular [1]. Moreover, there is empirical evidence that w G i ⊗ w H j , i.e., eigenvectors of L G◻H , are relatively close to eigenvectors of L G×H , i.e.,…”

Section: Estimating Laplacian Spectra Of Direct Product Graphsmentioning

confidence: 99%

Estimation of Laplacian spectra of direct and strong product graphs

Sayama

2016

Discrete Applied Mathematics

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Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have already been obtained analytically (mostly for adjacency spectra), characterization of Laplacian spectra of direct product and strong product graphs has remained an open problem. Here we develop practical methods to estimate Laplacian spectra of direct and strong product graphs from spectral properties of their factor graphs using a few heuristic assumptions. Numerical experiments showed that the proposed methods produced reasonable estimation with percentage errors confined within a 10% range for most eigenvalues. (C) 2015 Elsevier B.V. All rights reserved

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Section: Estimating Laplacian Spectra Of Direct Product Graphsmentioning

confidence: 99%

Estimation of Laplacian spectra of direct and strong product graphs

Sayama

2016

Discrete Applied Mathematics

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“…In particular, for k = 2 (the graph of Figure 2), it follows that [2] , (1) [8] , (−1) [16] , (−3) [4] , −11}.…”

Section: The Spectra and Laplacian Spectra Of Iterated Lexicographic mentioning

confidence: 99%

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Abreu

Cardoso

Carvalho

et al. 2017

Discrete Mathematics

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Consider two graphs G and H. Let H k [G] be the lexicographic product of H k and G, where H k is the lexicographic product of the graph H by itself k times. In this paper, we determine the spectrum of H k [G] and H k when G and H are regular and the Laplacian spectrum of H k [G] and H k for G and H arbitrary. Particular emphasis is given to the least eigenvalue of the adjacency matrix in the case of lexicographic powers of regular graphs, and to the algebraic connectivity and the largest Laplacian eigenvalues in the case of lexicographic powers of arbitrary graphs. This approach allows the determination of the spectrum (in case of regular graphs) and Laplacian spectrum (for arbitrary graphs) of huge graphs. As an example, the spectrum of the lexicographic power of the Petersen graph with the googol number (that is, 10 100 ) of vertices is determined. The paper finish with the extension of some well known spectral and combinatorial invariant properties of graphs to its lexicographic powers.

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“…Before we can proceed, we need to define the graph Laplacian L of product graphs as a relevant high-pass filter; its interpretation as an extension of the circulant graph Laplacian high-pass filter to higher dimensions, with associated property preservations, will be revisited in Sect 4.3. We note that the formation of L is not a reflection of the adjacency matrix relations, except in the case of the Cartesian product ( [51], [52]):…”

Section: Multi-dimensional Wavelet Analysis On Product Graphsmentioning

confidence: 99%

Splines and wavelets on circulant graphs

Kotzagiannidis

Dragotti

2019

Applied and Computational Harmonic Analysis

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We present novel families of wavelets and associated filterbanks for the analysis and representation of functions defined on circulant graphs. In this work, we leverage the inherent vanishing moment property of the circulant graph Laplacian operator, and by extension, the e-graph Laplacian, which is established as a parameterization of the former with respect to the degree per node, for the design of vertex-localized and critically-sampled higher-order graph (e-)spline wavelet filterbanks, which can reproduce and annihilate classes of (exponential) polynomial signals on circulant graphs. In addition, we discuss similarities and analogies of the detected properties and resulting constructions with splines and spline wavelets in the Euclidean domain. Ultimately, we consider generalizations to arbitrary graphs in the form of graph approximations, with focus on graph product decompositions. In particular, we proceed to show how the use of graph products facilitates a multi-dimensional extension of the proposed constructions and properties.

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On the Laplacian spectra of product graphs

Cited by 43 publications

References 24 publications

Estimation of Laplacian spectra of direct and strong product graphs

Estimation of Laplacian spectra of direct and strong product graphs

Spectra and Laplacian spectra of arbitrary powers of lexicographic products of graphs

Splines and wavelets on circulant graphs

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