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Generating Infinite Random Graphs

Part of: Combinatorial probability Graph theory

Published online by Cambridge University Press:  22 May 2018

Csaba Biró  and
Udayan B. Darji
Csaba Biró
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (csaba.biro@louisville.edu; ubdarj01@louisville.edu)
Udayan B. Darji*
Affiliation:
Department of Mathematics, University of Louisville, Louisville, KY 40292, USA (csaba.biro@louisville.edu; ubdarj01@louisville.edu)
*
*Corresponding author.
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Abstract

We define a growing model of random graphs. Given a sequence of non-negative integers {dn}n=0∞ with the property that di≤i, we construct a random graph on countably infinitely many vertices v0, v1… by the following process: vertex vi is connected to a subset of {v0, …, vi−1} of cardinality di chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graphs, and we also give probabilistic methods for constructing infinite random trees.

Keywords

Erdős–Rényi graphrandom graphinfinite graphtreeshomogeneous structure

MSC classification

Primary: 05C63: Infinite graphs 05C80: Random graphs
Secondary: 05C05: Trees 60C99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 847 - 868
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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