Metric Uniformization and Spectral Bounds for Graphs

Kelner, Jonathan A.; Lee, James R.; Price, Gregory N.; Teng, Shang‐Hua

doi:10.1007/s00039-011-0132-9

Cited by 24 publications

(34 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Theorem 1.2. Our results are tight and improve the recent results of Kelner, Lee, Price and Teng [17] on bounded genus graphs. In addition to providing a uniform arguably more conceptual proof of the results of [17,16,27], we hope that our method makes the above mentioned existing similarities between the methods used in the spectral theory of surfaces and graphs more transparent.…”

Section: Introductionsupporting

confidence: 90%

“…The above result also implies a similar bound for the eigenvalues of the standard Laplacian, at the expense of an extra d max factor. We note that Kelner, Lee, Price and Teng [17] give a similar bound for the standard spectrum with a linear rather than quadratic dependence in d max . However, their bound has a gk log(g + 1) 2 dependence instead of our (g + k) dependence.…”

Section: Introductionsupporting

confidence: 54%

“…Informally, the improvement over [17] means that the asymptotic behavior of graphs' eigenvalues do not depend on the (geometric) genus of the graph. This fact, which may be seen as a one-sided discrete form of Weyl's law for surfaces, is consistent with the intuition that at a small scale, bounded genus graphs behave like planar graphs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A transfer principle and applications to eigenvalue estimates for graphs

Amini

Cohen-Steiner

2018

Comment. Math. Helv.

View full text Add to dashboard Cite

In this paper, we prove a variant of the Burger-Brooks transfer principle which, combined with recent eigenvalue bounds for surfaces, allows to obtain upper bounds on the eigenvalues of graphs as a function of their genus. More precisely, we show the existence of a universal constants C such that the k-th eigenvalue λ nr k of the normalized Laplacian of a graph G of (geometric) genus g on n vertices satisfieswhere dmax denotes the maximum valence of vertices of the graph. This result is tight up to a change in the value of the constant C, and improves recent results of Kelner, Lee, Price and Teng on bounded genus graphs.To show that the transfer theorem might be of independent interest, we relate eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple graph models, and discuss an application to the mesh partitioning problem, extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang to arbitrary meshes.

show abstract

Section: Introductionsupporting

confidence: 90%

Section: Introductionsupporting

confidence: 54%

See 1 more Smart Citation

A transfer principle and applications to eigenvalue estimates for graphs

Amini

Cohen-Steiner

2018

Comment. Math. Helv.

View full text Add to dashboard Cite

show abstract

“…It has found many important applications in pure mathematics, see e.g. [26], [22], [25]. We discuss a result of that in this section which is needed in our arguments later.…”

Section: 2mentioning

confidence: 96%

Multi-way dual Cheeger constants and spectral bounds of graphs

Liu

2015

Advances in Mathematics

View full text Add to dashboard Cite

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.Comment: 30 pages, 1 figure, revised; Theorem 6.4 added. Comments are welcom

show abstract

“…For example, Kelner et al [KLPT11] show that for n-vertex, bounded-degree planar graphs, one has that the kth smallest eigenvalue satisfies λ k = O(k/n). However, to the best of our knowledge, universal lower bounds were known only for the second smallest eigenvalue of the normalized Laplacian.…”

Section: Related Workmentioning

confidence: 99%

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

Lyons

Gharan

2017

International Mathematics Research Notices

View full text Add to dashboard Cite

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R k . The primary use of this embedding has been for practical spectral clustering algorithms [SM00, NJW01]. Recently, spectral embedding was studied from a theoretical perspective to prove higher order variants of Cheeger's inequality [LOT12,LRTV12].We use spectral embedding to provide a unifying framework for bounding all the eigenvalues of graphs. For example, we show that for any finite connected graph with n vertices and all k ≥ 2, the kth largest eigenvalue is at most 1 − Ω(k 3 /n 3 ), which extends the only other such result known, which is for k = 2 only and is due to [LO81]. This upper bound improves to 1 − Ω(k 2 /n 2 ) if the graph is regular. We generalize these results, and we provide sharp bounds on the spectral measure of various classes of graphs, including vertex-transitive graphs and infinite graphs, in terms of specific graph parameters like the volume growth.As a consequence, using the entire spectrum, we provide (improved) upper bounds on the return probabilities and mixing time of random walks with considerably shorter and more direct proofs. Our work introduces spectral embedding as a new tool in analyzing reversible Markov chains. Furthermore, building on [Lyo05], we design a local algorithm to approximate the number of spanning trees of massive graphs.

show abstract

Metric Uniformization and Spectral Bounds for Graphs

Cited by 24 publications

References 52 publications

A transfer principle and applications to eigenvalue estimates for graphs

A transfer principle and applications to eigenvalue estimates for graphs

Multi-way dual Cheeger constants and spectral bounds of graphs

Sharp Bounds on Random Walk Eigenvalues via Spectral Embedding

Contact Info

Product

Resources

About