Journal ArticleDOI
Asymptotic enumeration by degree sequence of graphs with degrees o(n1/2)
Reads0
Chats0
TLDR
The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences to determine the asymptotic number of unlabelled graphs with a given degree sequence.Abstract:
We determine the asymptotic number of labelled graphs with a given degree sequence for the case where the maximum degree iso(|E(G)|1/3). The previously best enumeration, by the first author, required maximum degreeo(|E(G)|1/4). In particular, ifk=o(n
1/2), the number of regular graphs of degreek and ordern is asymptotically
$$\frac{{(nk)!}}{{(nk/2)!2^{nk/2} (k!)^n }}\exp \left( { - \frac{{k^2 - 1}}{4} - \frac{{k^3 }}{{12n}} + 0\left( {k^2 /n} \right)} \right).$$
Under slightly stronger conditions, we also determine the asymptotic number of unlabelled graphs with a given degree sequence. The method used is a switching argument recently used by us to uniformly generate random graphs with given degree sequences.read more
Citations
More filters
Book ChapterDOI
Surveys in Combinatorics, 1999: Models of Random Regular Graphs
TL;DR: This is a survey of results on properties of random regular graphs, together with an exposition of some of the main methods of obtaining these results.
MonographDOI
Introduction to random graphs
Alan Frieze,Michał Karoński +1 more
TL;DR: All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.
Journal ArticleDOI
Sudden Emergence of a Giantk-Core in a Random Graph
TL;DR: These proofs are based on the probabilistic analysis of an edge deletion algorithm that always find ak-core if the graph has one, and demonstrate that, unlike the 2-core, when ak- core appears for the first time it is very likely to be giant, of size ?pk(?k)n.
Journal ArticleDOI
A sequential importance sampling algorithm for generating random graphs with prescribed degrees
TL;DR: An extension of a combinatorial characterization due to Erdős and Gallai is used to develop a sequential algorithm for generating a random labeled graph with a given degree sequence, which allows for surprisingly efficient sequential importance sampling.
References
More filters
Journal ArticleDOI
Uniform generation of random regular graphs of moderate degree
TL;DR: This paper shows how to generate k -regular graphs on n vertices uniformly at random in expected time O ( nk 3), provided k = O(n 1 3 ) .
Journal ArticleDOI
Asymptotic enumeration by degree sequence of graphs of high degree
TL;DR: This work considers the estimation of the number of labelled simple graphs with degree sequence d 1, d 2, . . . , d n by using an n-dimensional Cauchy integral and gives as a corollary the asymptotic joint distribution function of the degrees of a random graph.
Journal ArticleDOI
Automorphisms of random graphs with specified vertices
TL;DR: Conditions are found under which the expected number of automorphisms of a large random labelled graph with a given degree sequence is close to 1, which produces an asymptotic formula for the number of unlabelledk-regular simple graphs onn vertices, as well as various asymPTotic results on the probable connectivity and girth of such graphs.