scholarly journals The number of graphs and a random graph with a given degree sequence

2012 ◽  
Vol 42 (3) ◽  
pp. 301-348 ◽  
Author(s):  
Alexander Barvinok ◽  
J.A. Hartigan
Keyword(s):  
Random Graph ◽  
Degree Sequence

scholarly journals A Note on the Warmth of Random Graphs with Given Expected Degrees

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Yilun Shang
Keyword(s):  
Statistical Mechanics ◽  
Chromatic Number ◽  
Random Graph ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Degree Sequence ◽  
Graph Model ◽  
Upper And Lower Bounds ◽  
Random Graph Model

We consider the random graph modelG(w)for a given expected degree sequencew=(w1,w2,…,wn). Warmth, introduced by Brightwell and Winkler in the context of combinatorial statistical mechanics, is a graph parameter related to lower bounds of chromatic number. We present new upper and lower bounds on warmth ofG(w). In particular, the minimum expected degree turns out to be an upper bound of warmth when it tends to infinity and the maximum expected degreem=O(nα)with0<α<1/2.

scholarly journals On the Degree Sequence of an Evolving Random Graph Process and Its Critical Phenomenon

2009 ◽  
Vol 46 (4) ◽  
pp. 1213-1220 ◽  
Author(s):  
Xian-Yuan Wu ◽  
Zhao Dong ◽  
Ke Liu ◽  
Kai-Yuan Cai
Keyword(s):  
Random Graph ◽  
Degree Distribution ◽  
Critical Phenomenon ◽  
Degree Sequence ◽  
Edge Deletion ◽  
Specific Parameter

In this paper we focus on the problem of the degree sequence for a random graph process with edge deletion. We prove that, while a specific parameter varies, the limit degree distribution of the model exhibits critical phenomenon.

scholarly journals Phase transition on the degree sequence of a random graph process with vertex copying and deletion

2011 ◽  
Vol 121 (4) ◽  
pp. 885-895 ◽  
Author(s):  
Kai-Yuan Cai ◽  
Zhao Dong ◽  
Ke Liu ◽  
Xian-Yuan Wu
Keyword(s):  
Phase Transition ◽  
Random Graph ◽  
Degree Sequence

scholarly journals The scaling window for a random graph with a given degree sequence

2012 ◽  
Vol 41 (1) ◽  
pp. 99-123 ◽  
Author(s):  
Hamed Hatami ◽  
Michael Molloy
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Scaling Window

scholarly journals How to determine if a random graph with a fixed degree sequence has a giant component

2017 ◽  
Vol 170 (1-2) ◽  
pp. 263-310 ◽  
Author(s):  
Felix Joos ◽  
Guillem Perarnau ◽  
Dieter Rautenbach ◽  
Bruce Reed
Keyword(s):  
Random Graph ◽  
Degree Sequence ◽  
Giant Component ◽  
Fixed Degree

The degree sequence of a random graph. I. The models

1997 ◽  
Vol 11 (2) ◽  
pp. 97-117 ◽  
Author(s):  
Brendan D. McKay ◽  
Nicholas C. Wormald
Keyword(s):  
Random Graph ◽  
Degree Sequence

scholarly journals Subgraph counts for dense random graphs with specified degrees

2020 ◽  
pp. 1-38
Author(s):  
Catherine Greenhill ◽  
Mikhail Isaev ◽  
Brendan D. McKay
Keyword(s):  
Random Graph ◽  
Random Graphs ◽  
Spanning Trees ◽  
Degree Sequence ◽  
Complex Function ◽  
Random Permutation ◽  
Expected Number ◽  
Asymptotic Expressions ◽  
Number Of Spanning Trees

Abstract We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d 1 , …, d n ) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes d j = λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

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