scholarly journals Fast uniform generation of random graphs with given degree sequences

2021 ◽  
Author(s):  
Andrii Arman ◽  
Pu Gao ◽  
Nicholas Wormald
Keyword(s):  
Random Graphs ◽  
Degree Sequences ◽  
Uniform Generation

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 Degree sequences of random graphs

1981 ◽  
Vol 33 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Béla Bollobás
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals The degree sequences and spectra of scale-free random graphs

2006 ◽  
Vol 29 (2) ◽  
pp. 226-242 ◽  
Author(s):  
Jonathan Jordan
Keyword(s):  
Random Graphs ◽  
Degree Sequences ◽  
Scale Free

scholarly journals Erratum: Quasi-random graphs with given degree sequences

2008 ◽  
Vol 33 (4) ◽  
pp. 536-536
Author(s):  
Fan Chung ◽  
Ron Graham
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals Quasi-random graphs with given degree sequences

2007 ◽  
Vol 32 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Fan Chung ◽  
Ron Graham
Keyword(s):  
Random Graphs ◽  
Degree Sequences

scholarly journals On the Spectra of General Random Graphs

10.37236/702 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Fan Chung ◽  
Mary Radcliffe
Keyword(s):  
Independent Random Variable ◽  
Random Graph ◽  
Random Graphs ◽  
Power Law ◽  
Error Bounds ◽  
Adjacency Matrix ◽  
Random Variable ◽  
Degree Sequences ◽  
Chernoff Inequality

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.

Connected Components in Random Graphs with Given Expected Degree Sequences

2002 ◽  
Vol 6 (2) ◽  
pp. 125-145 ◽  
Author(s):  
Fan Chung ◽  
Linyuan Lu
Keyword(s):  
Random Graphs ◽  
Connected Components ◽  
Degree Sequences

scholarly journals Resilient Degree Sequences with respect to Hamilton Cycles and Matchings in Random Graphs

10.37236/8279 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Padraig Condon ◽  
Alberto Espuny Díaz ◽  
Daniela Kühn ◽  
Deryk Osthus ◽  
Jaehoon Kim
Keyword(s):  
Random Graphs ◽  
Perfect Matching ◽  
Degree Sequence ◽  
Hamilton Cycle ◽  
Vertex Degree ◽  
Degree Sequences ◽  
Hamilton Cycles ◽  
Degree Condition ◽  
Degree Conditions

Pósa's theorem states that any graph $G$ whose degree sequence $d_1 \le \cdots \le d_n$ satisfies $d_i \ge i+1$ for all $i < n/2$ has a Hamilton cycle. This degree condition is best possible. We show that a similar result holds for suitable subgraphs $G$ of random graphs, i.e.~we prove a `resilience version' of Pósa's theorem: if $pn \ge C \log n$ and the $i$-th vertex degree (ordered increasingly) of $G \subseteq G_{n,p}$ is at least $(i+o(n))p$ for all $i<n/2$, then $G$ has a Hamilton cycle. This is essentially best possible and strengthens a resilience version of Dirac's theorem obtained by Lee and Sudakov. Chvátal's theorem generalises Pósa's theorem and characterises all degree sequences which ensure the existence of a Hamilton cycle. We show that a natural guess for a resilience version of Chvátal's theorem fails to be true. We formulate a conjecture which would repair this guess, and show that the corresponding degree conditions ensure the existence of a perfect matching in any subgraph of $G_{n,p}$ which satisfies these conditions. This provides an asymptotic characterisation of all degree sequences which resiliently guarantee the existence of a perfect matching.

Sign in / Sign up

Export Citation Format

Share Document