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Eigenvalues and expansion of regular graphs

1995
*
Journal of the ACM
*

We improve the lower bound ontheexpansion

doi:10.1145/210118.210136
fatcat:qjabzod7jrdjvmqdqnziwqye3y
*of*Ramanujan*graphs*to approximately k/2, Moreover. we construct afamilyof k-*regular**graphs*with asymptotically optimal second*eigenvalue**and*linear*expansion*... The spectral method yielded a lower bound*of*k\4 on the*expansion**of*Iinear-sized subsets*of*k-*regular*Ramanujan*graphs*. ... By repeating the construction in Theorem 5.2, we obtain k-*regular**graphs*whose second largest*eigenvalue*i~z absolute value is (2 + o(l))~~*and*linear*expansion*k/2. ...##
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Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders

2002
*
Annals of Mathematics
*

We are grateful to David Zuckerman for illuminating discussions

doi:10.2307/3062153
fatcat:af2vrucpcff2tf4rgcgcqfmfzq
*and*a number*of*useful suggestions early in the stages*of*this work. ... We thank the organizers*of*the DIMACS Workshop on Pseudorandomness*and*Explicit Combinatorial Constructions in ... It is easy to verify that this value is the largest*eigenvalue**of*the random walk on the infinite D-*regular*tree.*Eigenvalues**and**expansions*. ...##
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Eigenvalues and expanders

1986
*
Combinatorica
*

Here we show that a

doi:10.1007/bf02579166
fatcat:3jbj4ycbsrbtfd2ktjqa6bt45e
*regular*bipartite*graph*is an expander ifandonly if the second largest*eigenvalue**of*its adjacency matrix is well separated from the first. ... It also supplies an efficient algorithm for approximating the expanding properties*of*a*graph*. The exact determination*of*these properties is known to be coNP-complete. ... Added in proof: Recently, Lubotzky, Phillips,*and*Sarnak [34] constructed, for every fixed d=p+ 1, p prime, an infinite family*of*d-*regular**graphs*G with 2(G)>=d-2fd -1. ...##
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High-girth near-Ramanujan graphs with lossy vertex expansion
[article]

2021
*
arXiv
*
pre-print

For every d = p+1 for prime p

arXiv:2007.13630v2
fatcat:x6vokez4wrgjth3wcq6bqzqx5u
*and*infinitely many n, we exhibit an n-vertex d-*regular**graph*with girth Ω(log_d-1 n)*and*vertex*expansion**of*sublinear sized sets bounded by d+1/2 whose nontrivial*eigenvalues*... Kahale proved that linear sized sets in d-*regular*Ramanujan*graphs*have vertex*expansion*∼d/2*and*complemented this with construction*of*near-Ramanujan*graphs*with vertex*expansion*no better than d/2. ... Acknowledgements We would like to thank Shirshendu Ganguly*and*Nikhil Srivastava for their highly valuable insights, intuition,*and*comments. ...##
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Spectra of Random Regular Hypergraphs

2021
*
Electronic Journal of Combinatorics
*

We then relate the second

doi:10.37236/8741
fatcat:fehq5l7sxzc5rffc2sbaqmqqb4
*eigenvalues*to both its*expansion*property*and*the mixing rate*of*the non-backtracking random walk on*regular*hypergraphs. ... In this paper, we study the spectra*of**regular*hypergraphs following the definitions from Feng*and*Li (1996). ... Acknowledgments We thank Sebastian Cioabȃ*and*Kameron Decker Harris for helpful comments. This work was partially supported by NSF DMS-1949617. ...##
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Eigenvalues and expansion of bipartite graphs

2012
*
Designs, Codes and Cryptography
*

Finally we present a new bound on the

doi:10.1007/s10623-011-9598-6
fatcat:ghbndw6ytzfv5p4qyrxwnazn44
*expansion*coefficient*of*(c, d)-*regular*bipartite*graphs**and*compare that with with a classical bound. ... We prove lower bounds on the largest*and*second largest*eigenvalue**of*the adjacency matrix*of*connected bipartite*graphs**and*give necessary*and*sufficient conditions for equality. ... Finally we present a new bound on the*expansion*coefficient*of*(c, d)-*regular*bipartite*graphs**and*compare that with a classical bound. ...##
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Expansion in Matrix-Weighted Graphs
[article]

2020
*
arXiv
*
pre-print

There are natural generalizations

arXiv:2009.12008v1
fatcat:2aj2wlrkerbapesbs64uudqdf4
*of*the Laplacian*and*adjacency matrices for such*graphs*. These matrices can be used to define*and*control*expansion*for matrix-weighted*graphs*. ... A new definition*of*a matrix-weighted expander*graph*suggests the tantalizing possibility*of*families*of*matrix-weighted*graphs*with better-than-Ramanujan*expansion*. ... The adjacency*and*Laplacian spectra*of*a d-*regular*matrix weighted*graph*have related*eigenvalues*: since the total degree matrix D is equal to dI, the*eigenvalues**of*A are µ i = d − λ i . ...##
###
Spectra of random regular hypergraphs
[article]

2021
*
arXiv
*
pre-print

We then relate the second

arXiv:1905.06487v5
fatcat:i3bgw42xwbce7gjcsy36m52hnq
*eigenvalues*to both its*expansion*property*and*the mixing rate*of*the non-backtracking random walk on*regular*hypergraphs. ... In this paper, we study the spectra*of**regular*hypergraphs following the definitions from Feng*and*Li (1996). ... Acknowledgments We thank Sebastian Cioabȃ*and*Kameron Decker Harris for helpful comments. This work was partially supported by NSF DMS-1949617. ...##
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Entropy waves, the zig-zag graph product, and new constant-degree
[article]

2004
*
arXiv
*
pre-print

Taking a product

arXiv:math/0406038v1
fatcat:kocl3z2rnjalvoivhooup3n6gu
*of*a large*graph*with a small*graph*, the resulting*graph*inherits (roughly) its size from the large one, its degree from the small one,*and*its*expansion*properties from both! ... Subsequent work [ALW01], [MW01] relates the zig-zag product*of**graphs*to the standard semidirect product*of*groups, leading to new results*and*constructions on expanding Cayley*graphs*. ... We are grateful to David Zuckerman for illuminating discussions*and*a number*of*useful suggestions early in the stages*of*this work. ...##
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The Limiting Eigenvalue Distribution of Iterated k-Regular Graph Cylinders
[article]

2018
*
arXiv
*
pre-print

We explore the limiting empirical

arXiv:1807.07624v1
fatcat:roa3p2emmnfazd2mrfkkim3spe
*eigenvalue*distributions arising from matrices*of*the form A_n+1 = < b m a t r i x > , where A_0 is the adjacency matrix*of*a k-*regular**graph*. ... This research grew out*of*our work on neural networks in k-*regular**graphs*. ... The Distribution*of**Eigenvalues*Given a k-*regular**graph**and*its*graph*cylinder, iteration as shown in equation 2.2 produces a family*of**regular**graph*cylinders. ...##
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Formal Zeta Function Expansions and the Frequency of Ramanujan Graphs
[article]

2014
*
arXiv
*
pre-print

We then consider the expected value

arXiv:1406.4557v1
fatcat:n5adcpffebcmtkzxhoft63iqoq
*of*this power series over random, d-*regular**graph*on n vertices, with d fixed*and*n tending to infinity. ... Our formal computation has a natural analogue when we consider random covering*graphs**of*degree n over a fixed,*regular*"base*graph*." ... In more detail, we show that the logarithmic derivative*of*the Zeta function*of*a*regular**graph*has a simple power series*expansion*at infinity. ...##
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Large Low-Diameter Graphs are Good Expanders

2018
*
European Symposium on Algorithms
*

We revisit the classical question

doi:10.4230/lipics.esa.2018.71
dblp:conf/esa/DinitzSS18
fatcat:5zc7dgchovgkbaqgzrkxgjiphi
*of*the relationship between the diameter*of*a*graph**and*its*expansion*properties. ... A recurring theme is that the lower the diameter*of*the*graph**and*(more importantly) the larger its size, the better the*expansion*guarantees. ... Acknowledgements We wish to thank Nati Linial, Alex Samorodnitsky, Elchanan Mossel,*and*Noga Alon for fruitful discussions. ...##
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Design of highly synchronizable and robust networks

2010
*
Automatica
*

Golden spectral dynamical networks (

doi:10.1016/j.automatica.2010.06.046
fatcat:ujzpkqs6qvb7bketf2ay4s32y4
*graphs*) are those for which the spectral spread (the difference between the largest*and*smallest*eigenvalues**of*the adjacency matrix) is equal to the spectral gap (the ... In particular, the*regular*bipartite dynamical networks, reported here by the first time, have the best possible*expansion**and*consequently are the most robust ones against node/link failures or intentional ... SG thanks the support by the Ministry*of*Science*and*technology (Spain) under the project MTM2008-06620-C03-01/MTM. ...##
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On the extreme eigenvalues of regular graphs
[article]

2005
*
arXiv
*
pre-print

We also prove an analogue

arXiv:math/0407274v2
fatcat:4o2sq4akynca5okcpsjmuiaoje
*of*Serre's theorem regarding the least*eigenvalues**of*k-*regular**graphs*: given ϵ>0, there exist a positive constant c=c(ϵ,k)*and*a nonnegative integer g=g(ϵ,k) such that for any ... In this paper, we present an elementary proof*of*a theorem*of*Serre concerning the greatest*eigenvalues**of*k-*regular**graphs*. ... Analogous theorems for the least*eigenvalues**of**regular**graphs*The analogous result to Theorem 1 for the least*eigenvalues**of*a k-*regular**graph*is not true. ...##
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Spectral statistics for scaling quantum graphs
[article]

2006
*
arXiv
*
pre-print

The explicit solution to the spectral problem

arXiv:quant-ph/0608076v1
fatcat:o4vhpdb3mfefxfmu4ja3crd5wi
*of*quantum*graphs*is used to obtain the exact distributions*of*several spectral statistics, such as the oscillations*of*the quantum momentum*eigenvalues*around ... the average, δ k_n=k_n-k̅_n,*and*the nearest neighbor separations, s_n=k_n-k_n-1. ... SPECTRAL DISTRIBUTIONS FOR*REGULAR*QUANTUM*GRAPHS*In the simplest case*of*the so called*regular**graphs*[6, 7] the individual momentum*eigenvalues*can be expanded into a periodic orbit series, k n = π ...
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