Skip to main content
SpringerLink
Log in

Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions

  • Research
  • Published:
Research in Number Theory Aims and scope Submit manuscript

Abstract

For a prime \(p\equiv 3\pmod {4}\) and a positive integer t, let \(q=p^{2t}\). Let g be a primitive element of the finite field \(\mathbb {F}_q\). The Peisert graph \(P^*(q)\) is defined as the graph with vertex set \(\mathbb {F}_q\) where ab is an edge if and only if \(a-b\in \langle g^4\rangle \cup g\langle g^4\rangle \). We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in \(P^*(q)\). We also give a new proof for the number of complete subgraphs of order three contained in \(P^*(q)\) by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order m in Peisert graphs.

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

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Alexander, J.: Selected results in combinatorics and graph theory, Thesis (Ph.D.) University of Delaware (2016)

  2. Atanasov, R., Budden, M., Lambert, J., Murphy, K., Penland, A.: On certain induced subgraphs of Paley graphs. Acta Univ. Apulensis Math. Inform. 40, 51–65 (2014)

    Article  MathSciNet  Google Scholar 

  3. Barman, R., Kalita, G.: Hypergeometric functions over \(\mathbb{F} _q\) and traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 141(10), 3403–3410 (2013)

    Article  Google Scholar 

  4. Barman, R., Kalita, G.: Elliptic curves and special values of Gaussian hypergeometric series. J. Number Theory 133, 3099–3111 (2013)

    Article  MathSciNet  Google Scholar 

  5. Barman, R., Kalita, G.: On the polynomial \(x^d+ax+b\) over \(\mathbb{F} _q\) and Gaussian Hypergeometric Series. Int. J. Number Theory 9(7), 1753–1763 (2013)

    Article  MathSciNet  Google Scholar 

  6. Berndt, B.C., Williams, K.S., Evans, R.J.: Gauss and Jacobi sums, Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication. Wiley, New York (1998)

    Google Scholar 

  7. Bhowmik, A., Barman, R.: On a Paley-type graph on \(\mathbb{Z}_n\), Graphs and Combinatorics 38 (2022), no. 2, Paper No. 41, 25 pp

  8. Chao, C.: On the classification of symmetric graphs with a prime number of vertices. Trans. Am. Math. Soc. 158, 247–256 (1971)

    Article  MathSciNet  Google Scholar 

  9. Dawsey, M.L., McCarthy, D.: Generalized Paley graphs and their complete subgraphs of orders three and four. Res. Math. Sci. 8(2), Paper No. 18, 23 pp (2021)

  10. Erdős, P.: On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 7, 459–464 (1962)

    MathSciNet  Google Scholar 

  11. Evans, R.: Hypergeometric \({_{3}}F_2\) (1/4) evaluations over finite fields and Hecke eigenforms. Proc. Am. Math. Soc. 138(2), 517–531 (2010)

    Article  Google Scholar 

  12. Evans, R.J., Pulham, J.R., Sheehan, J.: On the number of complete subgraphs contained in certain graphs. J. Combin. Theory Ser. B 3, 364–371 (1981)

    Article  MathSciNet  Google Scholar 

  13. Frechette, S., Ono, K., Papanikolas, M.: Gaussian hypergeometric functions and traces of Hecke operators. Int. Math. Res. Not. 60, 3233–3262 (2004)

    Article  MathSciNet  Google Scholar 

  14. Fuselier, J.: Hypergeometric functions over \(\mathbb{F} _p\) and relations to elliptic curve and modular forms. Proc. Am. Math. Soc. 138(1), 109–123 (2010)

    Article  MathSciNet  Google Scholar 

  15. Goodman, A.W.: On sets of acquaintances and strangers at any party. Am. Math. Mon. 66, 778–783 (1959)

    Article  MathSciNet  Google Scholar 

  16. Greene, J.: Hypergeometric functions over finite fields. Trans. Am. Math. Soc. 301, 77–101 (1987)

    Article  MathSciNet  Google Scholar 

  17. Greene, J.: Character Sum Analogues for Hypergeometric and Generalized Hypergeometric Functions over Finite Fields. Ph.D. thesis, Univ. of Minnesota, Minneapolis, 1984

  18. Katre, S.A., Rajwade, A.R.: Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum. Math. Scand. 60, 52–62 (1987)

    Article  MathSciNet  Google Scholar 

  19. Lennon, C.: Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold. J. Number Theory 131(12), 2320–2351 (2011)

    Article  MathSciNet  Google Scholar 

  20. Lennon, C.: Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves. Proc. Am. Math. Soc. 139(6), 1931–1938 (2011)

    Article  MathSciNet  Google Scholar 

  21. Lim, T.K., Praeger, C.E.: On generalised Paley graphs and their automorphism groups. Michigan Math. J. 58, 293–308 (2009)

    MathSciNet  Google Scholar 

  22. McCarthy, D.: Transformations of well-poised hypergeometric functions over finite fields. Finite Fields Appl. 18(6), 1133–1147 (2012)

    Article  MathSciNet  Google Scholar 

  23. Ono, K.: Values of Gaussian hypergeometric series. Trans. Am. Math. Soc. 350(3), 1205–1223 (1998)

    Article  MathSciNet  Google Scholar 

  24. Ono, K.: The web of modularity: arithmetic of the coefficients of modular forms and \(q\)-series, CBMS Regional Conference Series in Mathematics, 102. Amer. Math. Soc, Providence, RI (2004)

  25. Paley, R.E.A.C.: On orthogonal matrices. J. Math. Phys. 12, 311–320 (1933)

    Article  Google Scholar 

  26. Peisert, W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001)

    Article  MathSciNet  Google Scholar 

  27. Thomason, A.G.: Partitions of Graphs. Ph.D. thesis, Cambridge University (1979)

  28. Vega, M.V.: Hypergeometric functions over finite fields and their relations to algebraic curves. Int. J. Number Theory 7(8), 2171–2195 (2011)

    Article  MathSciNet  Google Scholar 

  29. Wage, N.: Character sums and Ramsey properties of generalized Paley graphs. Integers 6, article number 18 (2006)

  30. Zhang, H.: Self-complementary symmetric graphs. J. Graph Theory 16, 1–5 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We are very grateful to both the referees for their valuable comments. We are extremely grateful to Ken Ono for previewing a preliminary version of this paper and for his helpful comments.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, North Guwahati, Guwahati, Assam, 781039, India

    Anwita Bhowmik & Rupam Barman

Authors
  1. Anwita Bhowmik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rupam Barman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anwita Bhowmik.

Ethics declarations

Conflicts of interest

The authors assert that there are no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bhowmik, A., Barman, R. Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions. Res. number theory 10, 26 (2024). https://doi.org/10.1007/s40993-024-00512-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40993-024-00512-x

Keywords

Mathematics Subject Classification

Search

Navigation