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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19 Jun 2017
TL;DR: An algorithm is presented that outputs a (1+ε)-spectral sparsifier of G with O(n/ε2) edges in Ο(m/εO(1)) time, based on a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references.
Abstract: For any undirected and weighted graph G=(V,E,w) with n vertices and m edges, we call a sparse subgraph H of G, with proper reweighting of the edges, a (1+Iµ)-spectral sparsifier if (1-e)xTLGx≤xT LH x≤(1+e)xTLGx holds for any xEℝn, where LG and LH are the respective Laplacian matrices of G and H. Noticing that Ω(m) time is needed for any algorithm to construct a spectral sparsifier and a spectral sparsifier of G requires Ω(n) edges, a natural question is to investigate, for any constant e, if a (1+e)-spectral sparsifier of G with O(n) edges can be constructed in Ο(m) time, where the Ο notation suppresses polylogarithmic factors. All previous constructions on spectral sparsification require either super-linear number of edges or m1+Ω(1) time. In this work we answer this question affirmatively by presenting an algorithm that, for any undirected graph G and e>0, outputs a (1+e)-spectral sparsifier of G with O(n/e2) edges in Ο(m/eO(1)) time. Our algorithm is based on three novel techniques: (1) a new potential function which is much easier to compute yet has similar guarantees as the potential functions used in previous references; (2) an efficient reduction from a two-sided spectral sparsifier to a one-sided spectral sparsifier; (3) constructing a one-sided spectral sparsifier by a semi-definite program.
57 citations
TL;DR: In this paper, the eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigen values of its generators.
Abstract: In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd.
57 citations
TL;DR: A graph is reconstructible if all but at most one of its eigenvalues are simple and have eigenvectors not orthogonal to c, where c is the vector with each entry equal to one.
57 citations
TL;DR: This work uses the leading eigenvector of a purported extremal graph to deduce structural properties about that graph and proves three conjectures regarding the maximization of spectral invariants over certain families of graphs.
57 citations
TL;DR: In this paper, the authors established relations between the energy of the line graph of a graph and the energy associated with the signless Laplacian matrices of the graph.
57 citations