Skip to main content
SpringerLink
Log in

On the Differential Polynomial of a Graph

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial \(B(G;x):={\sum}_{k=-n}^{\partial(G)}B_k(G)x^{n+k}\), where Bk(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G; x) and its coefficients. In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.

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. Akbari, S., Alikhani, S., Peng, Y. H.: Characterization of graphs using domination polynomials. European Journal of Combinatorics, 31(7), 1714–1724 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Armada, C. L., Canoy Jr, S. R.: A-Differential of Graphs. International Journal of Mathematical Analysis, 9(44), 2171–2180 (2015)

    Article  Google Scholar 

  3. Arocha, J. L., Llano, B.: Mean value for the matching and dominating polynomial. Discussiones Mathematicae Graph Theory, 20(1), 57–69 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Basilio, L. A., Bermudo, S., Leañnos, J., et al.: β-Differential of a Graph. Symmetry, 9(10), 205 (2017)

    Article  Google Scholar 

  5. Basilio, L. A., Bermudo, S., Sigarreta, J. M.: Bounds on the differential of a graph. Utilitas Mathematica, 103, 319–334 (2017)

    MathSciNet  MATH  Google Scholar 

  6. Beck, M., Blado, D., Crawford, J., et al.: On Weak Chromatic Polynomials of Mixed Graphs. Graphs and Combinatorics, 31(1), 91–98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bermudo, S.: On the Differential and Roman domination number of a graph with minimum degree two. Discrete Applied Mathematics, 232, 64–72 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bermudo, S., De la Torre, L., Martín-Caraballo, A. M., et al.: The differential of the strong product graphs. International Journal of Computer Mathematics, 92(6), 1124–1134 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bermudo, S., Fernau, H.: Lower bound on the differential of a graph. Discrete Mathematics, 312, 3236–3250 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bermudo, S., Fernau, H.: Computing the differential of a graph: hardness, approximability and exact algorithms. Discrete Applied Mathematics, 165, 69–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bermudo, S., Fernau, H.: Combinatorics for smaller kernels: the differential of a graph. Theoretical Computer Science, 562, 330–345 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bermudo, S., Fernau, H., Sigarreta, J. M.: The differential and the roman domination number of a graph. Applicable Analysis and Discrete Mathematics, 8(1), 155–171 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bermudo, S., Rodríguez, J. M., Sigarreta, J. M.: On the differential in graphs. Utilitas Mathematica, 97, 257–270 (2015)

    MathSciNet  MATH  Google Scholar 

  14. Birkhoff, G. D.: A determinant formula for the number of ways of coloring a map. The Annals of Mathematics, 2(14), 42–46 (1912)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brown, J. I., Dilcher, K., Nowakowski, R. J.: Roots of Independence Polynomials of Well Covered Graphs. Journal of Algebraic Combinatorics, 11(3), 197–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Alliance polynomial of regular graphs. Discrete Applied Mathematics, 225, 22–32 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. Carballosa, W., Hernández-Gómez, J. C., Rosario, O., et al.: Computing the strong alliance polynomial of a graph. Investigacion Operacional, 37(2), 115–123 (2016)

    MathSciNet  Google Scholar 

  18. Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Computing the alliance polynomial of a graph. Ars Combinatoria, 135, 163–185 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Distinctive power of the alliance polynomial for regular graphs. Electronic Notes in Discrete Mathematics, 46, 313–320 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Dohmen, K., Pönitz, A., Tittmann, P.: A new two-variable generalization of the chromatic polynomial. Discrete Mathematics and Theoretical Computer Science, 6, 69–90 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Farrell, E. J.: An introduction to matching polynomials. Journal of Combinatorial Theory, Series B, 27, 75–86 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Goddard, W., Henning, M. A.: Generalised domination and independence in graphs. Congressus Numerantium, 123, 161–171 (1997)

    MathSciNet  MATH  Google Scholar 

  23. Godsil, C. D., Gutman, I.: On the theory of the matching polynomial. Journal of Graph Theory, 5, 137–144 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. Gutman, I., Harary, F.: Generalizations of the matching polynomial. Utilitas Mathematica, 24, 97–106 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Hoede, C., Li, X.: Clique polynomials and independent set polynomials of graphs. Discrete Mathematics, 125, 219–228 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kotek, T., Makowsky, J. A.: Recurrence relations for graph polynomials on bi-iterative families of graphs. European Journal of Combinatorics, 41, 47–67 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Makowsky, J. A., Ravve, E. V., Blanchard, N. K.: On the location of roots of graph polynomials. European Journal of Combinatorics, 41, 1–19 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mashburn, J. L., Haynes, T. W., Hedetniemi, S. M., et al.: Differentials in graphs. Utilitas Mathematica, 69, 43–54 (2006)

    MathSciNet  MATH  Google Scholar 

  29. de Mier, A., Noy, M.: On graphs determined by their Tutte polynomials. Graphs and Combinatorics, 20(1), 105–119 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Noy, M.: On graphs determined by polynomial invariants. Theoretical Computer Science, 307(2), 365–384 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pushpam, P. R. L., Yokesh, D.: Differential in certain classes of graphs. Tamkang Journal of Mathematics, 41(2), 129–138 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Pushpam, P. R. L., Yokesh, D.: A-differentials and total domination in graphs. Journal of Discrete Mathematical Sciences and Cryptography, 16(1), 31–43 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  33. Read, R. C.: An introduction to chromatic polynomials. Journal of Combinatorial Theory, 4(1), 52–71 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sigarreta, J. M.: Differential in Cartesian Product Graphs. Ars Combinatoria, 126, 259–267 (2016)

    MathSciNet  MATH  Google Scholar 

  35. Slater, P. J.: Enclaveless sets and MK-systems. J. Res. Nat. Bur. Standards, 82(3), 197–202 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tittmann, P., Averbouch, I., Makowsky, J. A.: The enumeration of vertex induced subgraphs with respect to the number of components. European Journal of Combinatorics, 32(7), 954–974 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tutte, W. T.: A contribution to the theory of chromatic polynomials. Canad. J. Math., 6, 80–91 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zhang, C. Q.: Finding critical independent sets and critical vertex subsets are polynomial problems. SIAM Journal on Discrete Mathematics, 3(3), 431–438 (1990)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank the referees for a careful reading of the manuscript and for some helpful suggestions.

Author information

Authors and Affiliations

  1. Unidad Académica de Matemáticas, Universidad Autónoma de Zacatecas, Calzada Solidaridad entronque Paseo a la Bufa, 98060, Zacatecas, Zac., México

    Ludwin A. Basilio-Hernández & Jesús Leaños

  2. Department of Mathematics and Statistics, Florida International University, 11200 SW 8ht Street, Miami, Florida, 33199, USA

    Walter Carballosa

  3. Department of Mathematics, Miami Dade College, 300 NE Second Ave., Miami, FL 33132, Miami, Florida, USA

    Walter Carballosa

  4. Facultad de Matemáticas, Universidad Autónoma de Guerrero, Carlos E. Adame No.54 Col. Garita, 39650, Acalpulco Gro., México

    José M. Sigarreta

Authors
  1. Ludwin A. Basilio-Hernández
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Walter Carballosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jesús Leaños
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. José M. Sigarreta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ludwin A. Basilio-Hernández.

Additional information

The third was partially supported by PFCE-UAZ 2018–2019 grant. The last author was supported in part by two grants from Ministerio de Economía y Competitividad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Basilio-Hernández, L.A., Carballosa, W., Leaños, J. et al. On the Differential Polynomial of a Graph. Acta. Math. Sin.-English Ser. 35, 338–354 (2019). https://doi.org/10.1007/s10114-018-7307-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-018-7307-3

Keywords

MR(2010) Subject Classification

Search

Navigation