On topological properties of some convexpolytopes by using line operator on theirsubdivisions

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$(G),M_2$(G), Zagreb coindicesM_1$(G),M_2$(G),M_2$(G)hyper-Zagreb indexHM(G), atom-bond connectivity indexABC(G), sum connectivityindexχ(G)and geometric-arithmetic connectivity indexGA(G), by consideringGas linegraph of subdivision of some convex polytopes andGdenotes its complement.

PDF

___

[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, MATCHCommun. Math. Comput. Chem.52, 103–112, 2004.
[7]E. Estrada, L. Torres, L. Rodriguez and I. Gutman,An atom-bond connectivityindex, 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, NewYork, 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 quantitativestructure-property studies, J. Serb. Chem. Soc.65(8), 577–580, 2000.
[15]I. Gutman and N. Trinajstic,Graph theory and molecular orbitals. Totalπ-electronenergy 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 ́cand G. Popivoda,On Zagreb Indices andCoindices, 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 extremalgraphs 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 metricdimension, Util. Math.88, 43–57, 2012.
[21]M. Imran, A.Q. Baig and M.K. Shafiq,Classes of convex polytopes with constantmetric dimension, Util. Math.90, 85-99, 2013.
[22]A. Iranmanesh, I. Gutman, O. Khormali and A. Mahmiani,The edge versions of theWiener 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 graphof 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 ofsubdivision graphs of certain nanostructures, Appl. Math. Comput.273, 125–130,2016.
[26]M.F. Nadeem, S. Zafar and Z. Zahid,Some Topological Indices ofL(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, PureAppl. Chem.69, 1137–1152, 1997.
[29]D. Vukicevic and B. Furtula,Topological index based on the ratios of geometrical andarithmetical 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.