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1 results

  • 6 - Connectivity

  • from Part II - Erdős–Rényi–Gilbert Model
  • Alan Frieze, Carnegie Mellon University, Pennsylvania, Michał Karoński, Adam Mickiewicz University, Poznań, Poland
  • Book:
    Random Graphs and Networks: A First Course
    Published online:
    02 March 2023
    Print publication:
    09 March 2023, pp 78-84

  • 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.