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Eigenvalues and expansion of regular graphs

Nabil Kahale
1995 Journal of the ACM  
We improve the lower bound ontheexpansion 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.  ... 
doi:10.1145/210118.210136 fatcat:qjabzod7jrdjvmqdqnziwqye3y

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders

Omer Reingold, Salil Vadhan, Avi Wigderson
2002 Annals of Mathematics  
We are grateful to David Zuckerman for illuminating discussions 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.  ... 
doi:10.2307/3062153 fatcat:af2vrucpcff2tf4rgcgcqfmfzq

Eigenvalues and expanders

Noga Alon
1986 Combinatorica  
Here we show that a 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.  ... 
doi:10.1007/bf02579166 fatcat:3jbj4ycbsrbtfd2ktjqa6bt45e

High-girth near-Ramanujan graphs with lossy vertex expansion [article]

Theo McKenzie, Sidhanth Mohanty
2021 arXiv   pre-print
For every d = p+1 for prime p 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.  ... 
arXiv:2007.13630v2 fatcat:x6vokez4wrgjth3wcq6bqzqx5u

Spectra of Random Regular Hypergraphs

Ioana Dumitriu, Yizhe Zhu
2021 Electronic Journal of Combinatorics  
We then relate the second 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.  ... 
doi:10.37236/8741 fatcat:fehq5l7sxzc5rffc2sbaqmqqb4

Eigenvalues and expansion of bipartite graphs

Tom Høholdt, Heeralal Janwa
2012 Designs, Codes and Cryptography  
Finally we present a new bound on the 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.  ... 
doi:10.1007/s10623-011-9598-6 fatcat:ghbndw6ytzfv5p4qyrxwnazn44

Expansion in Matrix-Weighted Graphs [article]

Jakob Hansen
2020 arXiv   pre-print
There are natural generalizations 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 .  ... 
arXiv:2009.12008v1 fatcat:2aj2wlrkerbapesbs64uudqdf4

Spectra of random regular hypergraphs [article]

Ioana Dumitriu, Yizhe Zhu
2021 arXiv   pre-print
We then relate the second 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.  ... 
arXiv:1905.06487v5 fatcat:i3bgw42xwbce7gjcsy36m52hnq

Entropy waves, the zig-zag graph product, and new constant-degree [article]

Omer Reingold, Salil Vadhan, Avi Wigderson
2004 arXiv   pre-print
Taking a product 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.  ... 
arXiv:math/0406038v1 fatcat:kocl3z2rnjalvoivhooup3n6gu

The Limiting Eigenvalue Distribution of Iterated k-Regular Graph Cylinders [article]

Clark Alexander, Tara Nenninger, Danielle Tucker
2018 arXiv   pre-print
We explore the limiting empirical 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.  ... 
arXiv:1807.07624v1 fatcat:roa3p2emmnfazd2mrfkkim3spe

Formal Zeta Function Expansions and the Frequency of Ramanujan Graphs [article]

Joel Friedman
2014 arXiv   pre-print
We then consider the expected value 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.  ... 
arXiv:1406.4557v1 fatcat:n5adcpffebcmtkzxhoft63iqoq

Large Low-Diameter Graphs are Good Expanders

Michael Dinitz, Michael Schapira, Gal Shahaf, Michael Wagner
2018 European Symposium on Algorithms  
We revisit the classical question 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.  ... 
doi:10.4230/lipics.esa.2018.71 dblp:conf/esa/DinitzSS18 fatcat:5zc7dgchovgkbaqgzrkxgjiphi

Design of highly synchronizable and robust networks

Ernesto Estrada, Silvia Gago, Gilles Caporossi
2010 Automatica  
Golden spectral dynamical networks (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.  ... 
doi:10.1016/j.automatica.2010.06.046 fatcat:ujzpkqs6qvb7bketf2ay4s32y4

On the extreme eigenvalues of regular graphs [article]

Sebastian M. Cioaba
2005 arXiv   pre-print
We also prove an analogue 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.  ... 
arXiv:math/0407274v2 fatcat:4o2sq4akynca5okcpsjmuiaoje

Spectral statistics for scaling quantum graphs [article]

Yu. Dabaghian
2006 arXiv   pre-print
The explicit solution to the spectral problem 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  ... 
arXiv:quant-ph/0608076v1 fatcat:o4vhpdb3mfefxfmu4ja3crd5wi
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