scholarly journals Braess's paradox for the spectral gap in random graphs and delocalization of eigenvectors

2016 ◽  
Vol 50 (4) ◽  
pp. 584-611 ◽  
Author(s):  
Ronen Eldan ◽  
Miklós Z. Rácz ◽  
Tselil Schramm
Keyword(s):  
Random Graphs ◽  
Spectral Gap ◽  
Braess’S Paradox

On the Laplacian Eigenvalues of Gn,p

2007 ◽  
Vol 16 (6) ◽  
pp. 923-946 ◽  
Author(s):  
AMIN COJA-OGHLAN
Keyword(s):  
Random Graphs ◽  
Spectral Gap ◽  
Degree Sequences ◽  
Laplacian Eigenvalues ◽  
Combinatorial Laplacian ◽  
Laplacian Spectra

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian $\LL(\gnp)$ is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of $\LL(\core(G))$ is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of $\LL(\gnp)$ is 1-O (d−1/2) w.h.p.

scholarly journals Braess's Paradox in large random graphs

2010 ◽  
Vol 37 (4) ◽  
pp. 495-515 ◽  
Author(s):  
Gregory Valiant ◽  
Tim Roughgarden
Keyword(s):  
Random Graphs ◽  
Braess’S Paradox

scholarly journals Large Bounded Degree Trees in Expanding Graphs

10.37236/278 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
József Balogh ◽  
Béla Csaba ◽  
Martin Pei ◽  
Wojciech Samotij
Keyword(s):  
Random Graphs ◽  
Maximum Degree ◽  
Spectral Gap ◽  
Sufficient Condition ◽  
Regular Graphs ◽  
Bounded Degree ◽  
Large Trees ◽  
Remarkable Result ◽  
Bounded Degree Trees ◽  
Expanding Graphs

A remarkable result of Friedman and Pippenger gives a sufficient condition on the expansion properties of a graph to contain all small trees with bounded maximum degree. Haxell showed that under slightly stronger assumptions on the expansion rate, their technique allows one to find arbitrarily large trees with bounded maximum degree. Using a slightly weaker version of Haxell's result we prove that a certain family of expanding graphs, which includes very sparse random graphs and regular graphs with large enough spectral gap, contains all almost spanning bounded degree trees. This improves two recent tree-embedding results of Alon, Krivelevich and Sudakov.

scholarly journals The Spectral Gap of Random Graphs with Given Expected Degrees

10.37236/227 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Amin Coja-Oghlan ◽  
André Lanka
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Degree Distribution ◽  
High Probability ◽  
Spectral Gap ◽  
Sparse Graphs ◽  
Laplacian Eigenvalues

We investigate the Laplacian eigenvalues of a random graph $G(n,\vec d)$ with a given expected degree distribution $\vec d$. The main result is that w.h.p. $G(n,\vec d)$ has a large subgraph core$(G(n,\vec d))$ such that the spectral gap of the normalized Laplacian of core$(G(n,\vec d))$ is $\geq1-c_0\bar d_{\min}^{-1/2}$ with high probability; here $c_0>0$ is a constant, and $\bar d_{\min}$ signifies the minimum expected degree. The result in particular applies to sparse graphs with $\bar d_{\min}=O(1)$ as $n\rightarrow\infty$. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].

Random Graphs

1998 ◽  
Author(s):  
V. F. Kolchin
Keyword(s):  
Random Graphs
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