On Domination Topological Indices of Graphs

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A.M. Hanan Ahmed, Anwar Alwardi, M. Ruby Salestina, On Domination Topological Indices of Graphs, Int. J. Anal. Appl., 19 (1) (2021), 47-64.

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A.M. Hanan Ahmed, Anwar Alwardi, M. Ruby Salestina

Abstract

Topological indices and domination in graphs are the essential topics in the theory of graphs. A set of vertices D ⊆ V (G) is said to be a dominating set for G if any vertex v ∈ V - D is adjacent to some vertex u ∈ D. In this research work, we define a new degree of each vertex v ∈ V (G), called the domination degree of v and denoted by dd(v), along with this new degree some domination indices based on domination degree are introduced. We study some basic properties of the domination degree function. Exact values and bounds for domination Zagreb indices of some families of graphs including the join and corona product are obtained. Finally, we generalize the domination degree of the vertex and new general indices are defined.

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References

  1. A.R. Ashrafi, T. Doslic, A. Hamzeha, The Zagreb coindices of graph operations, Discrete Appl. Math. 158 (2010), 1571- 1578.
  2. T. Al-Fozan, P. Manuel, I. Rajasingh, R. Sundara Rajan, Computing Szeged Index of Certain Nanosheets Using Partition Technique, MATCH Commun. Math. Comput. Chem. 72 (2014), 339-353.
  3. M. Azari, A. Iranmanesh, Harary Index of Some Nano-Structures, MATCH Commun. Math. Comput. Chem. 71 (2014), 373-382.
  4. B. Basavanagoud, On minimal and vertex minimal dominating graph, J. Inform. Math. Sci. 1 (2009), 139-146.
  5. D. Dimitrov, On structural properties of trees with minimal atom-bond connectivity index II: Bounds on B 1 - and B 2 -branches, Discrete Appl. Math. 204 (2016), 90-116.
  6. T. Doslic, M. Hosseinzadeh, Eccentric connectivity polynomial of some graph operations, Utilitas Math. 84 (2011), 297-309.
  7. F.V. Fomin, F. Grandoni, A.V. Pyatkin, A.A. Stepanov. Combinatorial bounds via measure and conquer: bounding minimal dominating sets and applications. ACM Trans. Algorithms 5(1) (2008), 9.
  8. B. Furtula, I. Gutman, and M. Dehmer, On structure-sensitivity of degree-based topological indices, Appl. Math. Comput. 219 (2013), 8973-8978.
  9. I. Gutman, Degree-based topological indices, Croat. Chem. Acta, 86 (2013), 351””361.
  10. I. Gutman, K.C. Das, The first zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83-92.
  11. I. Gutman, B.Ruscic, N.Trinajstic, C.F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes, J. Chem. Phys. 62 (1975), 3399-3405.
  12. I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538.
  13. F. Harary, Graph Theory, Narosa Publishing House, New Delhi, 2001.
  14. T.W. Haynes, S.T. Hedetniemi, P.J.Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York. (1998).
  15. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, (eds.) Domination in Graphs: Advanced Topics. Marcel Dekker, Inc. New York. (1998).
  16. Ivan Gutman, Kragujevac Trees and Their Energy, Appl. Math. Inform. Mech. 2 (2014), 71-79.
  17. O. Ivanciuc, T.S. Balaban, and A.T. Balaban, Reciprocal distance matrix, related local vertex invariants and topological indices, J. Math. Chem. 12 (1993), 309-318.
  18. F.C. Jear, R. Letourneur, M. Liedloff, On the number of minimal dominating sets on some graph classes, Theor. Comput. Sci. 562 (2015), 634-642.
  19. B.N. Kavitha, I.P. Kelkar, K.R. Rajanna, Perfect Domination in Book Graph and Stacked Book Graph, Int. J. Math. Trends Technol. 56 (2018), 511-514.
  20. M.F. Nadeem, S. Zafar, Z. Zahid, On certain topological indices of the line graph of subdivision graphs, Appl. Math. Comput. 271 (2015), 790-794.
  21. P.S. Ranjini, V. Lokesha, The Smarandache-Zagreb Indices on the Three Graph Operators, Int. J. Math. Comb. 3 (2010), 1-10.
  22. V. Sharma, R. Goswami, A.K. Madan, Eccentric connectivity index: A novel highly discriminating topological descriptor for structure-property and structure-activity studies, J. Chem. Inf. Comput. Sci. 37 (1997), 273-282.
  23. M. Teresa de Bustos Munoz, J.L.G. Guirao, J. Vigo-Aguiar, Decomposition of pseudo-radioactive chemical products with a mathematical approach, J. Math. Chem. 52 (4) (2014), 1059-1065.
  24. H. Wiener, Structural determination of the paraffin boiling points, J. Amer. Chem. Soc. 69 (1947), 17-20.