On R degrees of vertices and R indices of graphs

  • Authors

    • Süleyman Ediz Faculty of Education, Yuzuncu Yil University, Van, Turkey
  • R degree, R indices, Topological indices, QSAR, QSPR.
  • Topological indices have been used to modeling biological and chemical properties of molecules in quantitive structure property relationship studies and quantitive structure activity studies. All the degree based topological indices have been defined via classical degree concept. In this paper we define a novel degree concept for a vertex of a simple connected graph: R degree. And also we define R indices of a simple connected graph by using the R degree concept. We compute the R indices for well-known simple connected graphs such as paths, stars, complete graphs and cycles.

  • References

    1. [1] Wiener H., 1947, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc., 69, 17-20. https://doi.org/10.1021/ja01193a005.

      [2] Gutman I., Trinajstić N., 1971, Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 17, 535-538. https://doi.org/10.1016/0009-2614(72)85099-1.

      [3] Gutman I., RuÅ¡Äić B., Trinajstić N., Wilcox C.N., 1975, Graph Theory and Molecular Orbitals. XII. Acyclic Polyenes, J. Chem. Phys. 62, 3399-3405. https://doi.org/10.1063/1.430994.

      [4] Randić M., 1975, on characterization of molecular branching, J. Amer. Chem. Soc. 97, 6609–6615. https://doi.org/10.1021/ja00856a001.

      [5] Cui Q., Zhong L., 2017, the general Randić index of trees with given number of pendent vertices. Appl. Math. Comput. 302, 111–121. https://doi.org/10.1016/j.amc.2017.01.021.

      [6] Chen Z., Su G., Volkmann L., 2017, Sufficient conditions on the zeroth-order general Randić index for maximally edge-connected graphs, Discrete Appl. Math. 218, 64–70. https://doi.org/10.1016/j.dam.2016.11.002.

      [7] GAO W., Jamil M.K., Farahani M.R., 2017, the hyper-Zagreb index and some graph operations. J. Appl. Math. Comput. 54, 263–275.

      [8] Ediz S., 2016, reduced second Zagreb index of bicyclic graphs with pendent vertices, Matematiche (Catania) 71, 135–147.

      [9] Ediz S., Maximum chemical trees of the second reverse Zagreb index. Pac. J. Appl. Math. 7, 287–291.

      [10] Hosamani S.M., Basavanagoud B., 2015, New upper bounds for the first Zagreb index. MATCH Commun. Math. Comput. Chem. 74, 97–101.

      [11] Vukicevic D., Sedlar J., Stevanovic D., 2017, Comparing Zagreb Indices for Almost All Graphs, MATCH Commun. Math. Comput. Chem. 78, no. 2, 323-336.

      [12] Bianchi M., Cornaro A., Palacios J. L., Torriero A., 2015, New bounds of degree–based topological indices for some classes of c-cyclic graphs, Discr. Appl. Math. 184, 62–75. https://doi.org/10.1016/j.dam.2014.10.037.

      [13] Das K. C., Xu K., Nam J., 2015, Zagreb indices of graphs, Front. Math. China 10, 567–582. https://doi.org/10.1007/s11464-015-0431-9.

      [14] Tache R. M., 2016, on degree–based topological indices for bicyclic graphs, MATCH Commun. Math. Comput. Chem. 76, 99–116.

      [15] Estrada E., Torres L., Rodríguez L., Gutman I., 1998, an atom-bond connectivity index: modelling the enthalpy of formation of alkanes. Indian J. Chem. 37A, 849–855.

      [16] Zhong L., Cui Q., 2015, on a relation between the atom bond connectivity and the first geometric arithmetic indices. Discrete Appl. Math., 185, 249–253. https://doi.org/10.1016/j.dam.2014.11.027.

      [17] Ashrafi A.R., Dehghan Z.T., Habibi N., 2015, Extremal atom bond connectivity index of cactus graphs. Commun. Korean Math. Soc. 30, 283–295. https://doi.org/10.4134/CKMS.2015.30.3.283.

      [18] Furtula B., 2016, Atom bond connectivity index versus Graovac Ghorbani analog. MATCH Commun. Math. Comput. Chem. 75, 233–242.

      [19] Dimitrov D., 2016, on structural properties of trees with minimal atom bond connectivity index II: Bounds on and branches. Discrete Appl. Math. 204, 90–116. https://doi.org/10.1016/j.dam.2015.10.010.

      [20] Zhang X.M., Yang Y., Wang H., Zhang X.D., 2016, Maximum atom bond connectivity index with given graph parameters. Discrete Appl. Math. 215, 208–217. https://doi.org/10.1016/j.dam.2016.06.021.

      [21] VukiÄević D., Furtula B., 2009, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges. J.Math. Chem. 46, 1369–1376. https://doi.org/10.1007/s10910-009-9520-x.

      [22] Yuan Y., Zhou B., Trinajstić N., 2010, on geometric arithmetic index. J. Math. Chem. 47, 833–841. https://doi.org/10.1007/s10910-009-9603-8.

      [23] Das K.C., 2010, On geometric arithmetic index of graphs. MATCH Commun. Math. Comput. Chem. 64, 619–630.

      [24] Raza Z., Bhatti A.A., Ali A., 2016, more on comparison between first geometric arithmetic index and atom bond connectivity index. Miskolc Math. Notes 17, 561–570. https://doi.org/10.18514/MMN.2016.1265.

      [25] Gao W., 2016, A note on general third geometric arithmetic index of special chemical molecular structures. Commun. Math. Res. 32, 131–141.

      [26] An M., Xiong L., Su G., 2016, the k ordinary generalized geometric arithmetic index. Util. Math. , 100; 383–405.

      [27] Zhong L., 2012, the harmonic index for graphs. Applied Mathematics Letters. 25, 561–566. https://doi.org/10.1016/j.aml.2011.09.059.

      [28] Li J., Lv J.B., Liu Y., 2016, the harmonic index of some graphs. Bull. Malays. Math. Sci. Soc. 39, 331–340. https://doi.org/10.1007/s40840-015-0289-8.

      [29] Ilić A., 2016, Note on the harmonic index of a graph. Ars Combin. 128, 295–299.

      [30] Zhou B., Trinajstić N., 2009, on a novel connectivity index. J. Math. Chem. 46, 1252–1270. https://doi.org/10.1007/s10910-008-9515-z.

      [31] Farahani M.R., 2015, Randić connectivity and sum connectivity indices for Capra designed of cycles. Pac. J. Appl. Math. 7, 11–17.

      [32] Akhter S., Imran M., Raza Z., 2016, on the general sum connectivity index and general Randić index of cacti. J. Inequal. Appl. 300. https://doi.org/10.1186/s13660-016-1250-6.

      [33] GAO W., Jamil M.K., Farahani M.R., 2017, the hyper-Zagreb index and some graph operations. J. Appl. Math. Comput. , 263–275.

      [34] Gutman I., Dehgardi N., Aram H., 2016, on general first Zagreb index of graphs with fixed maximum degree. Bull. Int. Math. Virtual Inst. 6,251–258.

      [35] Milovanović E. I., Bekakos P. M., Bekakos M. P., Milovanović I. Ž., 2017, Sharp bounds for the general Randić index R-1 of a graph. Rocky Mountain J. Math. 47, 259–266. https://doi.org/10.1216/RMJ-2017-47-1-259.

      [36] Mansour T., Rostami M.A., Elumalai S., Xavier B.A., 2017, Correcting a paper on the Randić and geometric-arithmetic indices. Turkish J. Math. 41, 27–32. https://doi.org/10.3906/mat-1510-115.

      [37] Glogić E., Zogić E., Glišović N., 2017, Remarks on the upper bound for the Randić energy of bipartite graphs. Discrete Appl. Math. 221, 67–70. https://doi.org/10.1016/j.dam.2016.12.005.

      [38] Liu B., Pavlović L.R., Divnić T., Tomica R.; Liu J., Stojanović M.M., 2013, on the conjecture of Aouchiche and Hansen about the Randić index. Discrete Math. 313, 225–235. https://doi.org/10.1016/j.disc.2012.10.012.

  • Downloads

  • How to Cite

    Ediz, S. (2017). On R degrees of vertices and R indices of graphs. International Journal of Advanced Chemistry, 5(2), 70-72. https://doi.org/10.14419/ijac.v5i2.7973