Advertisement

On the Differential Polynomial of a Graph

  • Ludwin A. Basilio-HernándezEmail author
  • Walter Carballosa
  • Jesús Leaños
  • José M. Sigarreta
Article
  • 20 Downloads

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.

Keywords

Graph polynomial differential of a graph 

MR(2010) Subject Classification

05C31 05C69 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

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

References

  1. [1]
    Akbari, S., Alikhani, S., Peng, Y. H.: Characterization of graphs using domination polynomials. European Journal of Combinatorics, 31(7), 1714–1724 (2010)MathSciNetzbMATHGoogle Scholar
  2. [2]
    Armada, C. L., Canoy Jr, S. R.: A-Differential of Graphs. International Journal of Mathematical Analysis, 9(44), 2171–2180 (2015)Google Scholar
  3. [3]
    Arocha, J. L., Llano, B.: Mean value for the matching and dominating polynomial. Discussiones Mathematicae Graph Theory, 20(1), 57–69 (2000)MathSciNetzbMATHGoogle Scholar
  4. [4]
    Basilio, L. A., Bermudo, S., Lea˜nos, J., et al.: β-Differential of a Graph. Symmetry, 9(10), 205 (2017)Google Scholar
  5. [5]
    Basilio, L. A., Bermudo, S., Sigarreta, J. M.: Bounds on the differential of a graph. Utilitas Mathematica, 103, 319–334 (2017)MathSciNetzbMATHGoogle Scholar
  6. [6]
    Beck, M., Blado, D., Crawford, J., et al.: On Weak Chromatic Polynomials of Mixed Graphs. Graphs and Combinatorics, 31(1), 91–98 (2015)MathSciNetzbMATHGoogle Scholar
  7. [7]
    Bermudo, S.: On the Differential and Roman domination number of a graph with minimum degree two. Discrete Applied Mathematics, 232, 64–72 (2017)MathSciNetzbMATHGoogle Scholar
  8. [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)MathSciNetzbMATHGoogle Scholar
  9. [9]
    Bermudo, S., Fernau, H.: Lower bound on the differential of a graph. Discrete Mathematics, 312, 3236–3250 (2012)MathSciNetzbMATHGoogle Scholar
  10. [10]
    Bermudo, S., Fernau, H.: Computing the differential of a graph: hardness, approximability and exact algorithms. Discrete Applied Mathematics, 165, 69–82 (2014)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Bermudo, S., Fernau, H.: Combinatorics for smaller kernels: the differential of a graph. Theoretical Computer Science, 562, 330–345 (2015)MathSciNetzbMATHGoogle Scholar
  12. [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)MathSciNetzbMATHGoogle Scholar
  13. [13]
    Bermudo, S., Rodríguez, J. M., Sigarreta, J. M.: On the differential in graphs. Utilitas Mathematica, 97, 257–270 (2015)MathSciNetzbMATHGoogle Scholar
  14. [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)MathSciNetzbMATHGoogle Scholar
  15. [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)MathSciNetzbMATHGoogle Scholar
  16. [16]
    Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Alliance polynomial of regular graphs. Discrete Applied Mathematics, 225, 22–32 (2017)MathSciNetzbMATHGoogle Scholar
  17. [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)MathSciNetGoogle Scholar
  18. [18]
    Carballosa, W., Rodríguez, J. M., Sigarreta, J. M., et al.: Computing the alliance polynomial of a graph. Ars Combinatoria, 135, 163–185 (2017)MathSciNetzbMATHGoogle Scholar
  19. [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)MathSciNetzbMATHGoogle Scholar
  20. [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)MathSciNetzbMATHGoogle Scholar
  21. [21]
    Farrell, E. J.: An introduction to matching polynomials. Journal of Combinatorial Theory, Series B, 27, 75–86 (1979)MathSciNetzbMATHGoogle Scholar
  22. [22]
    Goddard, W., Henning, M. A.: Generalised domination and independence in graphs. Congressus Numerantium, 123, 161–171 (1997)MathSciNetzbMATHGoogle Scholar
  23. [23]
    Godsil, C. D., Gutman, I.: On the theory of the matching polynomial. Journal of Graph Theory, 5, 137–144 (1981)MathSciNetzbMATHGoogle Scholar
  24. [24]
    Gutman, I., Harary, F.: Generalizations of the matching polynomial. Utilitas Mathematica, 24, 97–106 (1983)MathSciNetzbMATHGoogle Scholar
  25. [25]
    Hoede, C., Li, X.: Clique polynomials and independent set polynomials of graphs. Discrete Mathematics, 125, 219–228 (1994)MathSciNetzbMATHGoogle Scholar
  26. [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)MathSciNetzbMATHGoogle Scholar
  27. [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)MathSciNetzbMATHGoogle Scholar
  28. [28]
    Mashburn, J. L., Haynes, T. W., Hedetniemi, S. M., et al.: Differentials in graphs. Utilitas Mathematica, 69, 43–54 (2006)MathSciNetzbMATHGoogle Scholar
  29. [29]
    de Mier, A., Noy, M.: On graphs determined by their Tutte polynomials. Graphs and Combinatorics, 20(1), 105–119 (2004)MathSciNetzbMATHGoogle Scholar
  30. [30]
    Noy, M.: On graphs determined by polynomial invariants. Theoretical Computer Science, 307(2), 365–384 (2003)MathSciNetzbMATHGoogle Scholar
  31. [31]
    Pushpam, P. R. L., Yokesh, D.: Differential in certain classes of graphs. Tamkang Journal of Mathematics, 41(2), 129–138 (2010)MathSciNetzbMATHGoogle Scholar
  32. [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)MathSciNetzbMATHGoogle Scholar
  33. [33]
    Read, R. C.: An introduction to chromatic polynomials. Journal of Combinatorial Theory, 4(1), 52–71 (1968)MathSciNetzbMATHGoogle Scholar
  34. [34]
    Sigarreta, J. M.: Differential in Cartesian Product Graphs. Ars Combinatoria, 126, 259–267 (2016)MathSciNetzbMATHGoogle Scholar
  35. [35]
    Slater, P. J.: Enclaveless sets and MK-systems. J. Res. Nat. Bur. Standards, 82(3), 197–202 (1977)MathSciNetzbMATHGoogle Scholar
  36. [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)MathSciNetzbMATHGoogle Scholar
  37. [37]
    Tutte, W. T.: A contribution to the theory of chromatic polynomials. Canad. J. Math., 6, 80–91 (1954)MathSciNetzbMATHGoogle Scholar
  38. [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)MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ludwin A. Basilio-Hernández
    • 1
    Email author
  • Walter Carballosa
    • 2
    • 3
  • Jesús Leaños
    • 1
  • José M. Sigarreta
    • 4
  1. 1.Unidad Académica de MatemáticasUniversidad Autónoma de ZacatecasZacatecasMéxico
  2. 2.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
  3. 3.Department of MathematicsMiami Dade CollegeMiamiUSA
  4. 4.Facultad de MatemáticasUniversidad Autónoma de GuerreroAcalpulco Gro.México

Personalised recommendations