The diameter of connected components of random graphs

Abstract

This work investigates the probability distribution of the maximum of the diameters of the connected components of a random graph of the G_n,p model. (We call this maximum the depth d of the graph G_n,p). D is also defined as the maximum over all u,vεV of the quantities d(u,v) and 1, where d(u,v) is the length of the shortest path from u to v (if any) and +∞ otherwise. We prove that (1) there is a constant c>2 such that, for any probability p in the range $[0,{\text{ }}1] - [\frac{c}{n},{\text{ }}\frac{{2c}}{n} - (\frac{c}{n})^2 ]$, the graph G_n,p has average depth ↔d=0 (logn). Furthermore, the probability that d=0 (logn) tends to 1 as n tends to ∞. We also prove that for $p \geqslant \frac{c}{{\sqrt[3]{n}}}$ (where c>1 is a particular constant) the depth of G_n,p is less than or equal to 3 with probability tending to 1 as n tends to ∞. Although the results $\bar d = 0\left( {\log n} \right)$ can be deduced from results of [Erdös, Renyi, 60] for several values of p, the result for sparse graphs $(p = \theta (\frac{1}{n}))$ and for very dense graphs $(p \geqslant \frac{c}{{\sqrt[3]{n}}})$ are entirely new.

This research is partially supported by the NSF grants MCS-83-00630, DCR-85-03497 and by the Greek Ministry of Industry Energy and Technology.