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6 - Connectivity

from Part II - Erdős–Rényi–Gilbert Model

Published online by Cambridge University Press:  02 March 2023

Alan Frieze
Affiliation:
Carnegie Mellon University, Pennsylvania
Michał Karoński
Affiliation:
Adam Mickiewicz University, Poznań, Poland
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Summary

Whether a graph is connected, i.e., there is a path between any two of its vertices, is of particular importance. Therefore, in this chapter, we first establish the threshold for the connectivity of a random graph. We then view this property in terms of the graph process and show that w.h.p. the random graph becomes connected at precisely the time when the last isolated vertex joins the giant component. This “hitting time” result is the precursor to several similar results. After this, we deal with k-connectivity, i.e., the parameter that measures the strength of connectivity of a graph. We show that the threshold for this property is the same as for the existence of vertices of degree k in a random graph.

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Publisher: Cambridge University Press
Print publication year: 2023

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  • Connectivity
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book: Random Graphs and Networks: A First Course
  • Online publication: 02 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009260268.009
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  • Connectivity
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book: Random Graphs and Networks: A First Course
  • Online publication: 02 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009260268.009
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Connectivity
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book: Random Graphs and Networks: A First Course
  • Online publication: 02 March 2023
  • Chapter DOI: https://doi.org/10.1017/9781009260268.009
Available formats
×