# 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.

## Keywords

Random Graph Sparse Graph Dense Graph Balance Graph Biconnected Component## Preview

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## References

- Chin, F., J. Lam, and T. Chen, "Optimal Parallel Algorithms for the Connected Components Problem", CACM 25 (9), Sept. 1982, pp. 659–665.Google Scholar
- Chernoff, H., "A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations", Annals of Math. Statistics, 23 1952.Google Scholar
- Cook, S., "Towards a Complexity Theory of Synchronous Parallel Computations", Specker Symposium on Logic and Algorithms, Zurich, Feb. 5–11, 1980.Google Scholar
- Dekel, E., D. Nassimi, and S. Sahni, "Parallel Matrix and Graph Algorithms", SIAM J. Comp. 10 (4), Nov. 1981, pp. 657–675.CrossRefGoogle Scholar
- Dymond, P. and S.A. Cook, "Hardware Complexity and Parallel Computation", IEEE 21st Symposium on Foundations of Computer Science, 1980, pp. 360–372.Google Scholar
- Erdos, P. and A. Renyi, "On the Evolution of Random Graphs", Pub. Math. Inst. Hung. Acad. Sci. 5A, 1960, pp. 17–61, also "The Art of Counting", J. Spenser Editor, MIT Press.Google Scholar
- Feller W., "An Introduction to Probability Theory and Its Applications", Vol. 1, Third Edition, John Wiley and Sons, New York, 1968.Google Scholar
- Fortune, S. and J. Wyllie, "Parallelism in Random Access Machines", Proc. 10th ACM Symp. on Theory of Computing, May, 1978, pp. 114–118.Google Scholar
- Hirschberg, D., A. Chandra, and D. Sarwate, "Computing Connected Components on Parallel Computers", Commun. of the ACM 22 (8), Aug. 1979, pp. 461–464.Google Scholar
- Ja'Ja' J, "Graph Connectivity Problems on Parallel Computers", TR GS-78-05, Dept. of Computer Science, Penn. State Univ., PA, 1976.Google Scholar
- Nath, D. and S.N. Maheshwari, "Parallel Algorithms for the Connected Components and Minimal Spanning Tree Problems", Information Processing Letters, 14 (1), April 1982, pp. 7–11.CrossRefGoogle Scholar
- Reif, J. and P. Spirakis, "K-Connectivity in Random Undirected Graphs, "to appear in Journal of Discrete Mathematics, 1986.Google Scholar
- Savage, C. and J. Ja'Ja', "Fast, Efficient Parallel Algorithms for Some Graph Problems", SIAM J. Comp., 10 (4), Nov. 1981, pp. 682–691.CrossRefGoogle Scholar