...

  • Home
  • Articles
    • Current Issue
    • Archives
  • Authors
    • Author Guidelines
    • Policies
    • Downloads
  • Editors
  • Reviewers
...

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 68 | Issue 6 | Year 2022 | Article Id. IJMTT-V68I6P501 | DOI : https://doi.org/10.14445/22315373/IJMTT-V68I6P501

Domination Polynomials for Inverse Graphs


J. Baskar Babujee, Temesgen Engida Yimer
Received Revised Accepted Published
04 Apr 2022 23 May 2022 02 Jun 2022 12 Jun 2022
Abstract

Let ๐บ = (๐‘‰, ๐ธ) be a simple undirected graph with order n. The inverse, or complement, of a graph ๐บ is denoted by ๐บฬ…and is obtained by filling in all the remaining edges needed to form a complete graph while removing all previously existing edges. Equivalently, ๐‘‰ (๐บ) = ๐‘‰ (๐บฬ…) and E(๐บฬ…) = E(Kn) โˆ’ E(G) and also |๐ธ(๐บฬ…)| = ( ๐‘› 2 ) โˆ’ |๐ธ(๐บ)|. A domination polynomial of an inverse graph of G is given as ๐ท(๐บฬ…, ๐‘ฅ ) = โˆ‘ ๐‘‘(๐บฬ…, ๐‘ฅ )๐‘ฅ ๐‘› ๐‘– ๐‘–=๐›พ(๐บฬ…) , where ๐‘‘(๐บฬ…, ๐‘ฅ ) is the number of all dominating sets of ๐บฬ…with size ๐‘– and ฮณ(๐บฬ…) is the minimum domination number of ๐บฬ…. In this paper, we introduce a generating function called a domination polynomial for inverse dominating sets of cycle, star, wheel graphs, and triangular book graphs. Also, the bounds of the minimum dominating number ฮณ(G) discussed with their corresponding inverse of the minimum dominating number ฮณ(๐บฬ…).

Keywords
Dominating set, Domination polynomials, Inverse of graph.
References

๏ปฟ[1] Akbari, S. Alikhani and Y.H. Peng YH, Characterization of Graphs using Domination Polynomials, European J. Combin, 31(2010) 1714โ€“1724.
[2] S. Alikhani and Y.H. Peng, Dominating sets and Domination Polynomials of paths, Int. J. Math. Sci, 10(2009)1โ€“10.
[3] S. Alikhani and Y.H. Peng, Introduction to Domination Polynomial of a Graph, Ars Combin,114 (2014)257โ€“266.
[4] J.L. Arocha and B.Llano, Mean Value for the Matching and Dominating Polynomial, Discussiones Mathematicae Graph Theory, 20(1)(2000)57โ€“70
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Sater. Fundamentals of Domination of Graphs, Marcel Dekker, Inc, NewYork, (1989).
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Sater. Domination in Graphs:Advanced Topics, Marcel Dekker, Inc, NewYork, (1998).
[7] T. Kotek , J. Preen and P. Tittmann. Domination polynomials of graphs products, arXiv:305.1475v2[math.CO] (2013).
[8] P.S. Nair . Construction of self-complementary graphs, Discrete Math, 175 (1997) 283โ€“287.
[9] J. Peterson . Die Theorie der regularen Graphs, Acta Mathematica, 15(1) (1891)193-220.
[10] N.B. Rathod and K.K. Kanani. k-cordial Labeling of Triangular Book, Triangular Book with Book Mark and Jewel Graph, Global Journal of Pure and Applied Mathematics, 13(10) (2017) 6979โ€“6989.
[11] J.J. Sylvester . On an application of the New Atomic Theory to the Graphical Presentation of the Invariants and Covariant Of Binary Quantic, American Journal of Mathematics, 1 (1878) 161-228.
[12] Wyatt J, Desormeaux, and T.W. Haynes . Domination parameters of a graph and its complement, Discussiones mathematicae graph theory, 38 (2018)203โ€“215.
[13] T.E. Yimer and J. Baskar Babujee . Perfect Domination Polynomial of a Homogeneous Caterpillar Graph and Full Binary Tree, Mathematical Notes,111(2) (2022) 129โ€“136.
[14] T.E. Yimer and J. Baskar Babujee. Domination polynomial of a corona graph, IOSR Journal of Mathematics; 18(2) (2022) 47โ€“51.

Citation :

J. Baskar Babujee, Temesgen Engida Yimer, "Domination Polynomials for Inverse Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 68, no. 6, pp. 1-7, 2022. Crossref, https://doi.org/10.14445/22315373/IJMTT-V68I6P501

  • PDF
  • Abstract
  • Keywords
  • References
  • Citation
Abstract Keywords References Citation
  • Home
  • Authors Guidelines
  • Paper Submission
  • APC
  • Archives
  • Downloads
  • Open Access
  • Publication Ethics
  • Copyrights Infringement
  • Journals
  • FAQ
  • Contact Us

Follow Us

Copyright ยฉ 2025 Seventh Sense Research Groupยฎ . All Rights Reserved