# Subgraphs of random graphs with specified degrees

@inproceedings{McKay2011SubgraphsOR, title={Subgraphs of random graphs with specified degrees}, author={Brendan D. McKay}, year={2011} }

If a graph is chosen uniformly at random from all the graphs with a given degree sequence, what can be said about its subgraphs? The same can be asked of bipartite graphs, equivalently 0-1 matrices. These questions have been studied by many people. In this paper we provide a partial survey of the eld, with emphasis on two general techniques: the method of switchings and the multidimensional saddle-point method.

## 77 Citations

Subgraphs of Dense Random Graphs with Specified Degrees

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2011

This work proves some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph, using the multidimensional saddle-point method.

Induced subgraphs in sparse random graphs with given degree sequences

- Computer Science, MathematicsEur. J. Comb.
- 2012

The result is extended to the probability space of random graphs with a given degree sequence and gives an asymptotic formula for any d=o(n^1^/^3), provided that H does not contain almost all the edges of the random graph.

ASYMPTOTIC ENUMERATION OF GRAPHS WITH GIVEN DEGREE SEQUENCE

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We survey results on counting graphs with given degree sequence, focusing on asymptotic results, and mentioning some of the applications of these results. The main recent development is the proof of…

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- 2019

For a given random graph, a connected component that contains a finite fraction of the entire graph’s vertices is called giant. The study of these components started with the Erdős-Rényi model, where…

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

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2013

An asymptotic formula is obtained approximating the number of graphs within a relative error which approaches 0 as n grows and it is proved that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D.

Random dense bipartite graphs and directed graphs with specified degrees

- Mathematics, Computer ScienceRandom Struct. Algorithms
- 2009

The uniform distribution on the space of simple bipartite graphs with degree sequence S in one part and T in the other is studied; equivalently, binary matrices with row sums S and column sums T are studied.

Random dense bipartite graphs and directed graphs with specified degrees

- Mathematics
- 2009

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other;…

Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph

- Mathematics
- 2017

In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As…

Global eigenvalue fluctuations of random biregular bipartite graphs

- Mathematics
- 2020

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class of analytic functions. As a key step in the proof,…

Models of random regular graphs

- 2010

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. Related results on asymptotic enumeration are…

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An asymptotic formula is obtained approximating the number of graphs within a relative error which approaches 0 as n grows and it is proved that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D.

Random dense bipartite graphs and directed graphs with specified degrees

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The uniform distribution on the space of simple bipartite graphs with degree sequence S in one part and T in the other is studied; equivalently, binary matrices with row sums S and column sums T are studied.

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Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other;…

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