Skip to main content
SpringerLink
Log in

The Median of the Number of Simple Paths on Three Vertices in the Random Graph

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

We study the asymptotic behavior of the random variable equal to the number of simple paths on three vertices in the binomial random graph in which the edge probability equals the threshold probability of the appearance of such paths. We prove that, for any fixed nonnegative integer b and a sufficiently large number n of vertices of the graph, the probability that the number of simple paths on three vertices in the given random graph is b decreases with n. As a consequence of this result, we obtain the median of the number of simple paths on three vertices for sufficiently large n.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. B. Bollobás, Random Graphs, in Cambridge Stud. Adv. Math. (Cambridge Univ. Press, Cambridge, 2001), Vol. 73.

    Book  Google Scholar 

  2. A. N. Egorova and M. E. Zhukovskii, “Disproof zero-one law for existential monadic properties of a spare binomial random graph,” Dokl. Math. 99 (1), 68–70 (2019).

    Article  Google Scholar 

  3. S. Janson, T. Łuczak, and A. Ruciński, Random Graphs (Wiley-Interscience, New York, 2000).

    Book  Google Scholar 

  4. A. S. Razafimahatrata and M. Zhukovskii, “Zero-one laws for k-variable first-order logic of sparse random graphs,” Discrete Appl. Math. (2019) (in press).

  5. M. E. Zhukovskii and A. M. Raigorodskii, “Random graphs: models and asymptotic characteristics,” Uspekhi Mat. Nauk 70 (1 (421)), 33–81 (2015).

    MathSciNet  MATH  Google Scholar 

  6. M. E. Zhukovskii and A. M. Raigorodskii, Russian Math. Surveys 70 (1), 35–88 (2015).

    Article  MathSciNet  Google Scholar 

  7. S. Kargaltsev, D. Shabanov, and T. Shaikheeva, “Two values of the chromatic number of a sparse random graph,” Acta Math. Univ. Comenianae 88 (3), 849–854 (2019).

    MathSciNet  Google Scholar 

  8. A. Ruciński and A. Vince, “Balanced graphs and the problem of subgraphs of a random graph,” Congr. Numer. 49, 181–190 (1985).

    MathSciNet  MATH  Google Scholar 

  9. B. Bollobás, “Threshold functions for small subgraphs,” Math. Proc. Cambridge Philos. Soc. 90 (2), 197–206 (1981).

    Article  MathSciNet  Google Scholar 

  10. A. V. Burkin and M. E. Zhukovskii, “Small subgraphs and their extensions in a random distance graph,” Mat. Sb. 209 (2), 22–46 (2018).

    Article  MathSciNet  Google Scholar 

  11. A. V. Burkin and M. E. Zhukovskii, Sb. Math. 209 (2), 163–186 (2018).

    Article  MathSciNet  Google Scholar 

  12. S. D. Til’ga, “Distribution of small subgraphs in Buckley–Osthus random graphs,” Izv. Ross. Akad. Nauk Ser. Mat. 81 (2), 161–214 (2017).

    MathSciNet  Google Scholar 

  13. S. D. Til’ga, Izv. Math. 81 (2), 391–443 (2017).

    Article  MathSciNet  Google Scholar 

  14. J. H. Kim, B. Sudakov, and V. Vu, “Small subgraphs of random regular graphs,” Discrete Math. 307 (15), 1961–1967 (2007).

    Article  MathSciNet  Google Scholar 

  15. S. Kiselev and A. Kupavskii, “Sharp bounds for the chromatic number of random Kneser graphs,” Acta Math. Univ. Comenianae 88 (3), 861–865 (2019).

    MathSciNet  Google Scholar 

  16. K. P. Choi, “On the medians of Gamma distributions and an equation of Ramanujan,” Proc. Amer. Math. Soc. 121 (1), 245–251 (1994).

    Article  MathSciNet  Google Scholar 

  17. S. Ramanujan, “Some properties of Bernoulli’s numbers,” J. Indian Math. Soc. 3, 219–234 (1911).

    MATH  Google Scholar 

  18. G. Szegő, “Ű ber einige von S. Ramanujan Gestelte Aufgaben,” J. London Math. Soc. 3 (3), 225–232 (1928).

    Article  MathSciNet  Google Scholar 

  19. G. N. Watson, “Theorems stated by Ramanujan (V): Approximations connected with ex,” Proc. London Math. Soc. (2) 29 (4), 293–308 (1927).

    MATH  Google Scholar 

  20. K. Hamza, “The smallest uniform upper bound on the distance between the mean and the median of the binomial and Poisson distributions,” Statist. Probab. Lett. 23 (1), 21–25 (1995).

    Article  MathSciNet  Google Scholar 

  21. R. Kaas and J. M. Buhrman, “Mean, median, and mode in binomial distributions,” Statist. Neerlandica 34 (1), 13–18 (1980).

    Article  MathSciNet  Google Scholar 

  22. O. Riordan and L. Warnke, “The Janson inequalities for general up-sets,” Random Structures Algorithms 46 (2), 391–395 (2015).

    Article  MathSciNet  Google Scholar 

  23. W. Feller, “On the normal approximation to the binomial distribution,” Ann. Math. Statistics 16, 319–329 (1945).

    Article  MathSciNet  Google Scholar 

  24. B. C. Berndt and R. A. Rankin, Ramanujan: Letters and Commentary (Amer. Math. Soc., Providence, RI, 1995).

    Book  Google Scholar 

  25. P. Flajolet, P. J. Grabner, P. Kirschenhofer, and H. Prodinger, “On Ramanujan’s Q-function,” J. Comput. Appl. Math. 58 (1), 103–116 (1995).

    Article  MathSciNet  Google Scholar 

  26. S. E. Alm, “Monotonicity of the difference between median and mean of gamma distributions and a related Ramanujan sequence,” Bernoulli 9 (2), 351–371 (2003).

    Article  MathSciNet  Google Scholar 

  27. H. Alzer, “On Ramanujan’s inequalities for exp(k),” J. London Math. Soc. (2) 69 (3), 639–656 (2004).

    Article  MathSciNet  Google Scholar 

  28. K. Jodgeo and S. M. Samuels, “Monotone convergence of binomial probabilities and a generalization Ramanujan’s equation,” Ann. of Math. Stat. 39 (4), 1191–1195 (1968).

    Article  Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation under grant 16-11-10014.

Author information

Authors and Affiliations

  1. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, 141701, Russia

    M. E. Zhukovskii

Authors
  1. M. E. Zhukovskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. E. Zhukovskii.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 1, pp. 49–58.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhukovskii, M.E. The Median of the Number of Simple Paths on Three Vertices in the Random Graph. Math Notes 107, 54–62 (2020). https://doi.org/10.1134/S000143462001006X

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S000143462001006X

Keywords

Search

Navigation