On topological properties of some convex polytopes by using line operator on their subdivisions

Yıl 2020, Cilt: 49 Sayı: 1, 136 - 146, 06.02.2020

Fatima Asif Zohaib Zahid Sohail Zafar Mohammad R. Farahani , Wei Gao

https://doi.org/10.15672/HJMS.2019.671

Cited By: 3

Öz

In this paper, we give theoretical results for some topological indices such as Zagreb indices $M_{1}(G)$, $M_{2}(G)$, $M_{3}(G)$, $R(G)$, $M_{1}(\overline{G})$, $M_{2}(\overline{G})$, Zagreb coindices $\overline{M_{1}}(G)$, $\overline{M_{2}}(G)$, $\overline{M_{2}}(\overline{G})$ hyper-Zagreb index $HM(G)$,atom-bond connectivity index $ABC(G)$, sum connectivity index $\chi(G)$ and geometric-arithmetic connectivity index $GA(G)$, by considering $G$ as line graph of subdivision of some convex polytopes and $\overline{G}$ denotes its complement.

Anahtar Kelimeler

Topological indices, line graph, subdivision, convex polytopes

Kaynakça

• [1] M.O. Alberton, The irregularity of a graph, Ars Combin. 46, 219–225, 1997.
• [2] M. Baca, Labellings of two classes of convex polytopes, Util. Math. 34, 24–31, 1988.
• [3] M. Baca, On magic labellings of convex polytopes, Ann. Discrete Math. 51, 13–16, 1992.
• [4] S.H. Bertz, The bond graph, J. C. S. Chem. Commun. 818–820, 1981.
• [5] B. Bollobas and P. Erdös, Graphs of extremal weights, Ars Combin. 50, 225-233, 1998.
• [6] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52, 103–112, 2004.
• [7] E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An atom-bond connectivity index, Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37, 849–855, 1998.
• [8] G.H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 65, 79–84, 2011.
• [9] B. Grünbaum, Graduate text in mathematics convex polytopes, Springer-Verlag, New York, 2003.
• [10] I. Gutman, Selected properties of the schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34, 1087–1089, 1994.
• [11] I. Gutman, Edge versions of topological indices, in: Novel Molecular Structure De- scriptors - Theory and Applications II , Univ. Kragujevac, Kragujevac, 2010.
• [12] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 83–92, 2004.
• [13] I. Gutman and E. Estrada, Topological indices based on the line graph of the molec- ular graph, J. Chem. Inf. Comput. Sci. 36, 541–543, 1996.
• [14] I. Gutman and Z. Tomovic, On the application of line graphs in quantitative structure-property studies, J. Serb. Chem. Soc. 65 (8), 577–580, 2000.
• [15] I. Gutman and N. Trinajstic, Graph theory and molecular orbitals. Total -electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17, 535–538, 1972.
• [16] I. Gutman, B. Furtula, A.A. Toropov and A.P. Toropova, The graph of atomic or- bitals and its basic properties. 2. Zagreb indices, MATCH Commun. Math. Comput. Chem. 53, 225–230, 2005.
• [17] I. Gutman, B. Furtula, Z.K. Vuki´cevi´c and G. Popivoda, On Zagreb Indices and Coindices, MATCH Commun. Math. Comput. Chem. 74, 5–16, 2015.
• [18] I. Gutman, L. Popovic, B.K. Mishra, M. Kaunar, E. Estrada and N. Guevara, Ap- plication of line graphs in physical chemistry. Predicting surface tension of alkanes, J. Serb. Chem. Soc. 62, 1025–1029, 1997.
• [19] P. Hansen, H. Melot and I. Gutman, Variable neighborhood search for extremal graphs 12. A note on the variance of bounded degrees in graphs, MATCH Commun. Math. Comput. Chem. 54, 221–232, 2005.
• [20] M. Imran, A.Q. Baig and A. Ahmed, Families of plane graphs with constant metric dimension, Util. Math. 88, 43–57, 2012.
• [21] M. Imran, A.Q. Baig and M.K. Shafiq, Classes of convex polytopes with constant metric dimension, Util. Math. 90, 85-99, 2013.
• [22] A. Iranmanesh, I. Gutman, O. Khormali and A. Mahmiani, The edge versions of the Wiener index, MATCH Comm. Math. Comput. Chem. 61, 663–672, 2009.
• [23] M. Randic, On Characterization of Molecular Branching, J. Amer. Chem. Soc. 97, 6609–6615, 1975.
• [24] M.F. Nadeem, S. Zafar and Z. Zahid, On certain Topological indices of the line graph of subdivision graphs, Appl. Math. Comput. 271, 790–794, 2015.
• [25] M.F. Nadeem, S. Zafar and Z. Zahid, On Topological properties of the line graphs of subdivision graphs of certain nanostructures, Appl. Math. Comput. 273, 125–130, 2016.
• [26] M.F. Nadeem, S. Zafar and Z. Zahid, Some Topological Indices of L(S(CNCk[n]), Punjab Univ. J. Math. (Lahore), 49 (1), 13–17, 2017.
• [27] G.H. Shirdel, H. Rezapour and A.M. Sayadi, The hyper-Zagreb index of graph oper- ations, Iran. J. Math. Chem. 4 (2), 213–220, 2013.
• [28] H. Van de Waterbeemd, R.E. Carter, G. Grassy, H. Kubiny, Y.C. Martin, M.S. Tutte, and P. Willet, Glossary of terms used in computational drug design, Pure Appl. Chem. 69, 1137–1152, 1997.
• [29] D. Vukicevic and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges, J. Math. Chem. 46, 1369–1376, 2009.
• [30] B. Zhou, Zagreb indices, MATCH Commun. Math. Comput. Chem. 52, 113–118, 2004.
• [31] B. Zhou and N. Trinajstic, On general sum-connectivity index, J. Math. Chem. 47, 210–218, 2010.