The common minimal equitable dominating signed graphs

P. Siva Kota Reddy and U. K. Misra
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 18, 2012, Number 4, Pages 40–46
Full paper (PDF, 150 Kb)

Details

Authors and affiliations

P. Siva Kota Reddy
Department of Mathematics, Acharya Institute of Technology
Soladevanahalli, Bangalore-560 090, India
* Corresponding author

U. K. Misra
Department of Mathematics, Berhampur University
Berhampur-760 007, Orissa, India

Abstract

In this paper, we define the common minimal equitable dominating signed graph of a given signed graph and offer a structural characterization of common minimal equitable dominating signed graphs. In the sequel, we also obtained switching equivalence characterization: \overline \Sigma ~ CMED(Σ), where \overline \Sigma are CMED(Σ) are complementary signed graph and common minimal equitable dominating signed graph of Σ respectively

Keywords

  • Signed graphs
  • Balance
  • Switching
  • Complement
  • Common minimal equitable dominating signed graph
  • Negation

AMS Classification

  • 05C22

References

  1. Abelson, R. P., M. J. Rosenberg, Symbolic psychologic: A model of attitudinal cognition, Behav. Sci., Vol. 3, 1958, 1–13.
  2. Berge, C., Theory of Graphs and its Applications, Methuen, London, 1962.
  3. Cockayne, E. J., S. T. Hedetniemi, Towards a theory of domination in graphs, Networks,Vol. 7, 1977, 247–261.
  4. De Jaenisch, C. F., Applications de l’Analyse mathematique an Jen des Echecs, 1862.
  5. Deepak, G., N. D. Soner, A. Alwardi, Vertex minimal and common minimal equitable dominating graphs, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, No. 10, 499–505.
  6. Easley, D., J. Kleinberg, Networks, Crowds, and Markets: Reasoning About a Highly Connected World, Cambridge University Press, 2010.
  7. Harary, F., Graph Theory, Addison-Wesley Publishing Co., 1969.
  8. Harary, F., On the notion of balance of a signed graph, Michigan Math. J., Vol. 2, 1953,143–146.
  9. Harary, F., Structural duality, Behav. Sci., Vol. 2, 1957, No. 4, 255–265.
  10. Ore, O., Theory of Graphs. Amer. Math. Soc. Colloq. Publ., Vol. 38, 1962.
  11. Rangarajan, R., P. Siva Kota Reddy, The edge C4 signed graph of a signed graph, Southeast Asian Bulletin of Mathematics, Vol. 34, 2010, No. 6, 1077–1082.
  12. Rangarajan, R., M. S. Subramanya, P. Siva Kota Reddy, Neighborhood signed graphs, Southeast Asian Bulletin of Mathematics, Vol. 36, 2012, No. 3, 389–397.
  13. Rouse Ball,W.W., Mathematical Recreation and Problems of Past and Present Times, 1892.
  14. Sampathkumar, E., Point signed and line signed graphs, Nat. Acad. Sci. Letters, Vol. 7, 1984, No. 3, 91–93.
  15. Sampathkumar, E., P. Siva Kota Reddy, M. S. Subramanya, Directionally n-signed graphs, Ramanujan Math. Soc., Lecture Notes Series (Proc. Int. Conf. ICDM 2008), Vol. 13, 2010, 155–162.
  16. Sampathkumar, E., P. Siva Kota Reddy, M. S. Subramanya, Directionally n-signed graphs-II, International J. Math. Combin., Vol. 4, 2009, 89–98.
  17. Sampathkumar, E., M. S. Subramanya, P. Siva Kota Reddy, Characterization of line sidigraphs, Southeast Asian Bulletin of Mathematics, Vol. 35, 2011, No. 2, 297–304.
  18. Siva Kota Reddy, P., M. S. Subramanya, Note on path signed graphs. Notes on Number Theory and Discrete Mathematics, Vol. 15, 2009, No. 4, 1–6.
  19. Siva Kota Reddy, P., S. Vijay, V. Lokesha, nth Power signed graphs, Proceedings of the Jangjeon Math. Soc., Vol. 12, 2009, No. 3, 307–313.
  20. Siva Kota Reddy, P., t-Path Sigraphs, Tamsui Oxford J. of Math. Sciences, Vol. 26, 2010, No. 4, 433–441.
  21. Siva Kota Reddy, P., E. Sampathkumar, M. S. Subramanya, Common-edge signed graph of a signed graph, J. Indones. Math. Soc., Vol. 16, 2010, No. 2, 105–112.
  22. Siva Kota Reddy, P., B. Prashanth, T. R. Vasanth Kumar, Antipodal signed directed Graphs, Advn. Stud. Contemp. Math., Vol. 21, 2011, No. 4, 355–360.
  23. Siva Kota Reddy, P., B. Prashanth, The Common Minimal Dominating Signed Graph, Trans. Comb., Vol. 1, 2012, No. 3, 39–46.
  24. Siva Kota Reddy, P., B. Prashanth, S-Antipodal signed graphs, Tamsui Oxford J. of Inf. Math. Sciences, Vol. 28, 2012, No. 2, 165–174.
  25. Siva Kota Reddy, P., S. Vijay, The super line signed graph Lr(S) of a signed Graph, Southeast Asian Bulletin of Mathematics, Vol. 36, 2012, No. 6, 875–882
  26. Siva Kota Reddy, P.,d B. Prashanth, Note on Minimal Dominating Signed Graphs, Bull. of Pure & Appl. Math., Vol. 7, 2013, No. 1, to appear.
  27. Sozánsky, T., Enueration of weak isomorphism classes of signed graphs, J. Graph Theory, Vol. 4, 1980, No. 2, 127–144.
  28. Yaglom, A. M., I. M. Yaglom. Challenging mathematical problems with elementary solutions. Volume 1: Combinatorial Analysis and Probability Theory, 1964.
  29. Zaslavsky, T., Signed graphs, Discrete Appl. Math., Vol. 4, 1982, No. 1, 47–74.
  30. Zaslavsky, T., A mathematical bibliography of signed and gain graphs and allied areas, VIII Edition, Electron. J. Combin., Dynamic Surveys in Combinatorics (1998), No. DS8. Eighth ed. (2012).

Related papers

Cite this paper

Siva Kota Reddy, P. & Misra, U. (2012). The common minimal equitable dominating signed graphs. Notes on Number Theory and Discrete Mathematics, 18(4), 40-46.

Comments are closed.