# The diameter of connected components of random graphs

• Paul Spirakis
Randomness Considerations
Part of the Lecture Notes in Computer Science book series (LNCS, volume 246)

## Abstract

This work investigates the probability distribution of the maximum of the diameters of the connected components of a random graph of the Gn,p model. (We call this maximum the depth d of the graph Gn,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 Gn,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 Gn,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.

## Keywords

Random Graph Sparse Graph Dense Graph Balance Graph Biconnected Component
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1987

## Authors and Affiliations

• Paul Spirakis
• 1
• 2
1. 1.Courant Institute of Mathematical Sciences, NYUUSA
2. 2.Computer Technology InstituteGreece