The Pure and Applied Mathematics (한국수학교육학회지시리즈B:순수및응용수학)

Korean Society of Mathematical Education (한국수학교육학회)
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EXTREMAL CHEMICAL TREES WITH RESPECT TO HYPER-ZAGREB INDEX

  • Ghalavand, Ali (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan) ;
  • Ashrafi, Ali Reza (Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan) ;
  • Sharafdini, Reza (Department of Mathematics, Faculty of Science, Persian Gulf University) ;
  • Ori, Ottorino (Actinum Chemical Research)
  • Received : 2018.11.28
  • Accepted : 2019.05.28
  • Published : 2019.08.31
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Abstract

Suppose G is a molecular graph with edge set E(G). The hyper-Zagreb index of G is defined as $HM(G)={\sum}_{uv{\in}E(G)}[deg_G(u)+deg_G(v)]^2$, where $deg_G(u)$ is the degree of a vertex u in G. In this paper, all chemical trees of order $n{\geq}12$ with the first twenty smallest hyper-Zagreb index are characterized.

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References

  1. A. T. Balaban & I. Motoc & D. Bonchev & O. Mekenyan: Topological indices for structure activity correlations. Topics Curr. Chem. 114 (1983), 21-55. https://doi.org/10.1007/BFb0111212
  2. B. Basavanagoud & V.R. Desai: Forgotten Topological Index and Hyper-Zagreb Index of Generalized Transformation Graphs. Bull. Math. Sci. Appl. 14 (2016), 1-6.
  3. B. Basavanagoud & S. Patil: A note on Hyper-Zagreb index of graph operations. Iran. J. Math. Chem. 7 (2016), no. 1, 89-92.
  4. C. Bey: An upper bound on the sum of squares of degrees in a hypergraph. Discrete Math. 269 (2003), 259-263. https://doi.org/10.1016/S0012-365X(03)00130-4
  5. K.C. Das: Sharp bounds for the sum of the squares of the degrees of a graph. Kragujevac J. Math. 25 (2003), 31-49.
  6. D. de Caen: An upper bound on the sum of squares of degrees in a graph. Discrete Math. 185 (1998), 245-248. https://doi.org/10.1016/S0012-365X(97)00213-6
  7. J. Devillers & A.T. Balaban: Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach Science Publishers, 1999.
  8. M. Eliasi & A. Ghalavand: Extremal trees with respect to some versions of Zagreb indices via majorization. Iranian J. Math. Chem. 8 (2017), no. 4, 391-401.
  9. S. Elumalai, T. Mansour & M.A. Rostami: New bounds on the hyper-Zagreb index for the simple connected graphs. Electronic Journal of Graph Theory and Applications 6 (2018), no. 1, 166-177. https://doi.org/10.5614/ejgta.2018.6.1.12
  10. F. Falahati-Nezhad & M. Azari: Bounds on the hyper-Zagreb index. J. Appl. Math. & Informatics 34 (2016), 319-330. https://doi.org/10.14317/jami.2016.319
  11. M. Karelson: Molecular Descriptors in QSAR/QSPR. Wiley, New York, 2000.
  12. I. Gutman & K.C. Das: The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92.
  13. I. Gutman: Graphs with smallest sum of squares of vertex degree. Kragujevac J. Math. 25 (2003), 51-54
  14. I. Gutman & N. Trinajstic: Graph theory and molecular orbital. Total ${\pi}$-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17 (1972), 535-538. https://doi.org/10.1016/0009-2614(72)85099-1
  15. I. Gutman: On hyper-Zagreb index and coindex. Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.) 42 (2017), 1-8.
  16. S. Nikolic, G. Kovacevic, A. Milicevic & N. Trinajstic: The Zagreb indices 30 years after. Croat. Chem. Acta 76 (2003), 113-124.
  17. K. Pattabiraman & M. Vijayaragavan: Hyper Zagreb indices and its coindices of graphs. Bull. Int. Math. Virt. Inst. 7 (2017), 31-41.
  18. H. Rezapoura, R. Nasiri & S. Mousavi: The hyper-Zagreb index of trees and unicyclic graphs. arXiv:1808.10441 [math.CO].
  19. G.H. Shirdel, H. Rezapour & A.M. Sayadi: The hyper-Zagreb index of graph operations. Iran. J. Math. Chem. 4 (2013), no. 2, 213-220.
  20. R. Todeschini & V. Consonni: Handbook of Molecular Descriptors. Wiley, VCH, 2000.
  21. M. Veylaki, M.V. Nikmehr & H.A. Tavallaee: The third and hyper-Zagreb coindices of some graph operations. J. Appl. Math. Compt. 50 (2016), 315-325. https://doi.org/10.1007/s12190-015-0872-z
  22. B. Zhou & N. Trinajstic: On general sum-connectivity index. J. Math. Chem. 47 (2010), 210-218. https://doi.org/10.1007/s10910-009-9542-4