ON CERTAIN TOPOLOGICAL INDICES OF THE DERIVED GRAPHS OF SUBDIVISION GRAPHS

Year 2016, Volume: 6 Issue: 2, 324 - 332, 01.12.2016

S. M. Hosamani V. Lokesha I. N. Cangul K. M. Devendraiah

Abstract

The derived graph [G]y of a graph G is the graph having the same vertex set as G, with two vertices of [G]y being adjacent if and only if their distance in G is two. Topological indices are valuable in the study of QSAR/QSPR. There are numerous applications of graph theory in the eld of structural chemistry. In this paper, we compute generalized Randic, general Zagreb, general sum-connectivity, ABC, GA; ABC4, and GA5 indices of the derived graphs of subdivision graphs.

Keywords

topological indices , line graph , subdivision graph , derived graph.

References

• Ashrafi,A.R., Doˇsli´c,T., and Hamzeh,A., (2010), The Zagreb coindices of graph operations, Discrete Appl. Math., 158, pp. 1571–1578.
• Basavanagoud,B., Gutman,I. and Gali,C.S., (2015), On second Zagreb index and coindex of some derived graphs, Kragujevac J. Sci., 37, pp. 113–121.
• Doˇsli´c,T., (2008), Vertex-weighted Wiener polynomials for composite graphs, Ars Mathematica Con- tempooranea, 1, pp. 66–80.
• Estrada,E., Torres,L., Rodriguez,L. and Gutman,I., (1998), An atom-bond connectivity index: Mod- elling the enthalpy of formation of alkanes. Indian J. Chem., 37A, pp. 849–855.
• Furtula,B., Graovac,A. and Vukiˇcevi´c,D., (2010), Augmented Zagreb index. Journal of Mathematical Chemistry., 48, pp. 370–380.
• Ghorbani,M. and HosseinzadehM.A., (2010), Computing ABC4index of nanostar dendrimers, Opto- electron. Adv. Mater.-Rapid Commun., 4(9), pp. 1419–1422.
• Graovac,A., Ghorbani,M. and Hosseinzadeh,M.A., Computing fifth geometric-arithmetic index for nanostar dendrimers., J. Math. Nanosci, 1, pp. 33–42.
• Gutman,I. and Trinajsti´c,N., (1972), Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17, pp. 535–538.
• Gutman,I., (1994), Selected properties of the Schultz molecular topological index, J. Chem. Inf. Com- put. Sci., 34, pp. 1087–1089.
• Gutman,I., Furtula,B. and Elphick,C., (2014), Three new/old vertexdegreebased topological indices, MATCH Commun. Math. Comput. Chem., 72, pp. 617–632.
• Hande,S.P., Jog,S.R., Ramane,H.S., Hampiholi,P.R., Gutman,I. and Durgi,B.S., (2013), Derived graphs of subdivision graphs, Kragujevac J. Sci., 37(2), pp. 319–323.
• Harary,F., (1969), Graph Theory, Addison–Wesely, Reading.
• Hosamani,S.M. and Gutman,I., (2014), Zagreb indices of transformation graphs and total transfor- mation graphs, Appl. Math. Comput., 247, pp. 1156-1160.
• Hosamani,S.M. and Basavanagoud,B., (2015), New upper bounds for the first Zagreb index, MATCH Commun. Math. Comput. Chem., 74(1), pp. 97–101.
• Li,X. and Shi,Y., (2008), A survey on the Randi´c index, MATCH Commun. Math. Comput. Chem., 59(1), pp. 127-156.
• Li,X. and Zhao,H., (2004), Trees with the first three smallest and largest generalized topological indices, MATCH Commun. Math. Comput. Chem., 50, pp. 57-62.
• Nadeem,M.F., Zafar,S. and Zahid,Z., (2015), Certain topological indicies of the line graph of subdiv- sion graphs, Appl. Math. Comput., (271), pp. 790–794.
• Randi´c,M., (1974), On characterization of molecular branching, J. Am. Chem. Soc., 97, pp. 6609–6615. [19] Ranjini,P.S., Lokesha,V. and Cangul,I.N., (2011), On the Zagreb indices of the line graphs of the subdivision graphs, Appl. Math. Comput., 218, pp. 699–702.
• Ranjini,P.S., Lokesha,V. and Rajan,M.A., (2011), On the Shultz index of the subdivision graphs, Adv. Stud. Contemp. Math., 21(3), pp. 279-290.
• Shirdel,G.H., Rezapour,H. and Sayadi,A.M., (2013), The hyper-Zagreb index of graph operations, Iran. J. Math. Chem., 4(2), pp. 213–220.
• Su,G. and Xu,L., (2015), Topological indices of the line graph of subdivision graphs and their Schur- bounds, Appl. Math. Comput., 253, pp. 395–401.
• Todeschini,R. and Consonni,V., (2000), Handbook of Molecular Descriptors, Wiley-VCH, Weinheim. [24] Vukicevic,D. and Furtula,B., (2009), Topological index based on the ratios of geometrical and arith- metical means of end-vertex degrees of edges, J. Math. Chem., 46, pp. 1369–1376.
• Yu,G. and Feng,L., (2013), On connective eccentricity index of graphs, MATCH Commun. Math. Comput. Chem., 69, pp. 611-628.
• Zhou,B. and Trinajsti´c,N., (2010), On general sum-connectivity index, J. Math. Chem., 47, pp. 210– 218.