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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.


Papers
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Proceedings ArticleDOI
13 May 2013
TL;DR: In this paper, the Dirichlet spectral gap of the truncated graph Laplacian is computed for the graphs of a dozen communication networks at the IP-layer, which are subgraphs of the much larger global IP-Layer network.
Abstract: Good clustering can provide critical insight into potential locations where congestion in a network may occur. A natural measure of congestion for a collection of nodes in a graph is its Cheeger ratio, defined as the ratio of the size of its boundary to its volume. Spectral methods provide effective means to estimate the smallest Cheeger ratio via the spectral gap of the graph Laplacian. Here, we compute the spectral gap of the truncated graph Laplacian, with the so-called Dirichlet boundary condition, for the graphs of a dozen communication networks at the IP-layer, which are subgraphs of the much larger global IP-layer network. We show that i) the Dirichlet spectral gap of these networks is substantially larger than the standard spectral gap and is therefore a better indicator of the true expansion properties of the graph, ii) unlike the standard spectral gap, the Dirichlet spectral gaps of progressively larger subgraphs converge to that of the global network, thus allowing properties of the global network to be efficiently obtained from them, and (iii) the (first two) eigenvectors of the Dirichlet graph Laplacian can be used for spectral clustering with arguably better results than standard spectral clustering. We first demonstrate these results analytically for finite regular trees. We then perform spectral clustering on the IP-layer networks using Dirichlet eigenvectors and show that it yields cuts near the network core, thus creating genuine single-component clusters. This is much better than traditional spectral clustering where several disjoint fragments near the network periphery are liable to be misleadingly classified as a single cluster. Since congestion in communication networks is known to peak at the core due to large-scale curvature and geometry, identification of core congestion and its localization are important steps in analysis and improved engineering of networks. Thus, spectral clustering with Dirichlet boundary condition is seen to be more effective at finding bona-fide bottlenecks and congestion than standard spectral clustering.

13 citations

Proceedings ArticleDOI
TL;DR: This paper justifies the use of the graph Laplacian's eigenbasis as a surrogate for the Fourier basis for graphs, and establishes an analogous uncertainty principle relating the two quantities, showing the degree to which a function can be simultaneously localized in the graph and spectral domains.
Abstract: The classical uncertainty principle provides a fundamental tradeoff in the localization of a function in the time and frequency domains. In this paper we extend this classical result to functions defined on graphs. We justify the use of the graph Laplacian's eigenbasis as a surrogate for the Fourier basis for graphs, and define the notions of "spread" in the graph and spectral domains. We then establish an analogous uncertainty principle relating the two quantities, showing the degree to which a function can be simultaneously localized in the graph and spectral domains.

13 citations

Journal ArticleDOI
Hajime Urakawa1
TL;DR: A new approach to estimate the Cheeger constant, the heat kernel, and the Green kernel of the combinatorial Laplacian for an infinite graph is given.
Abstract: This paper gives a new approach to estimate the Cheeger constant, the heat kernel, and the Green kernel of the combinatorial Laplacian for an infinite graph.

13 citations

Journal ArticleDOI
TL;DR: In this article, the Laplacian spectra of a 3-prism graph with planar and polyhedral structure were calculated and applied to calculate the number of spanning trees and mean first-passage time.
Abstract: In this paper, we calculate the Laplacian spectra of a 3-prism graph and apply them. This graph is both planar and polyhedral, and belongs to the generalized Petersen graph. Using the regular structures of this graph, we obtain the recurrent relationships for Laplacian matrix between this graph and its initial state — a triangle — and further derive the corresponding relationships for Laplacian eigenvalues between them. By these relationships, we obtain the analytical expressions for the product and the sum of the reciprocals of all nonzero Laplacian eigenvalues. Finally we apply these expressions to calculate the number of spanning trees and mean first-passage time (MFPT) and see that the scaling of MFPT with the network size N is N2, which is larger than those performed on some uniformly recursive trees.

13 citations

Journal ArticleDOI
TL;DR: In this article, the authors derived necessary spectral conditions for the existence of graph homomorphisms in which they also consider some parameters related to the corresponding eigenspaces such as nodal domains.

13 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
202316
202236
202153
202086
201981