STATUS CONNECTIVITY INDICES OF CARTESIAN PRODUCT OF GRAPHS

In this paper, we establish one of the recent topological indices called the first status connectivity index S1 G = P uv2E G [G u + G v ] and second status connectivity index S2 G = P uv2E G [G u G v ] of Cartesian product of two simple graphs are determined. Also these indices are computed for nanotube, nanotorus, grid and cartesian product of complete graphs.

___

  • [1] Adiga, C. and Malpashree, R., (2016), The degree status connectivity index of graphs and its multiplicative version, South Asian J. of Math., 6(6), pp. 288 - 299.
  • [2] Ashrafi, A.R. and Ghorbani, M., (2010), Eccentric connectivity index of fullerenes In: I. Gutman, B. Furtula,(eds.) Novel Molecular Structure Descriptors - Theory and Applications II, Uni. Kragujevac, Kragujevac, pp. 183 - 192.
  • [3] Ashrafi, A.R., Saheli, M. and Ghorbani, M., (2011), The eccentric connectivity index of nanotubes and nanotori, J. Comput. Appl. Math., 235, pp. 4561 - 4566.
  • [4] Balaban, A.T., (1976), Chemical Applications of Graph Theory, Academic Press, London.
  • [5] Das, K.C., Xu, K. and Nam, J., (2015), Zagreb indices of graphs, Front. Math. China, 10, pp. 567 - 582.
  • [6] Das, K.C., Lee, D. and Graovac, A., (2013), Some properties of the Zagreb eccentricity indices, Ars Math. Contemp., 6, pp. 117 - 125.
  • [7] Devillers, J. and Balaban, A.T., (1999), Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach, Amsterdam, The Netherlands.
  • [8] Furtula, B., Gutman, I. and Dehmer, M., (2013), On structure sensitivity of degree based topological indices, Appl. Math. Comput., 219, pp. 8973 - 8978.
  • [9] Graovac, A., Gutman, I. and Vukiˇcevi´c, D. (Ed.), (2009), Mathematical Methods and Modelling for Students of Chemistry and Biology, Hum Press, Zagreb.
  • [10] Gutman, I. and Trinajstic, N., (1972), Graph theory and molecular orbitals, Total Π electron energy of alternate hydrocarbons, Chem. Phy. Letters, 17, pp. 535 - 538.
  • [11] Gutman, I., (2003), Introduction to Chemical Graph Theory, Fac. Sci. Kragujevac, Kragujevac (in Serbian).
  • [12] Gutman, I.(Ed.), (2006), Mathematical Methods in Chemistry, Prijepolje Museum, Prijepolje.
  • [13] Gutman, I. and Das, K.C., (2004), The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 50, pp. 83 - 92.
  • [14] Gutman, I., (2013), Degree based topological indices, Croat. Chem. Acta, 86, pp. 351 - 361.
  • [15] Gutman, I. and Furtula, B., (2015), Metric extremal graphs in: Dehmer, M. and Emmert-Streib. F. (Eds.), Quantitative Graph Theory Mathematical Foundations and Applications, CRC Press, Boca Raton, pp. 111 - 139.
  • [16] Harary, F., (1959), Status and contrastatus, Sociometry, 22, pp. 23 - 43.
  • [17] Illi´c, A. and Milosavljevi´c, N., (2013), The Weighted vertex PI index, Math. Comput. Model, 57, pp. 623 - 631.
  • [18] Illi´c, A. and Gutman, I., (2011), Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem., 65, pp. 731 - 744.
  • [19] Khalifeh, M.H., Yousefi-Azari, H. and Ashrafi, A.R., (2009), The first and second Zagreb indices of some graph operations, Discrete Appl. Math., 157, pp. 804 - 8
  • [20] Klavˇzar, S., (2007), On the PI index: PI-partitions and Cartesian product graphs, MATCH Commun. Math. Comput. Chem., 57, pp. 573-586.
  • [21] Nilanjan, De., Abu Nayeem, Sk. Md. and Anita, P., (2016), The F-coindex of some graph operations, Springer Plus, 5 (221), 13 pages.
  • [22] Pattabiraman, K. and Kandan, P., (2016), On weighted PI index of graphs, Elect. Notes in Discrete Math., 53, pp. 225 - 238.
  • [23] Pattabiraman, K. and Kandan, P., (2015), Generalization on degree distance of tensor product of graphs, Aust. J. Combin., 62, pp. 211 - 227.
  • [24] Pattabiraman, K. and Kandan, P., (2015), Generalization on degree distance of strong product of graphs, Iranian J. Math. Sci.& informatics, 10(2), pp. 87 - 98.
  • [25] Pattabiraman, K. and Kandan, P., (2014), Weighted PI index of corona product of graphs, Discrete Math. Alg. Appl., 6(4), 1450055 (9 pages).
  • [26] Pattabiraman, K. and Paulraja, P., (2012), Wiener and vertex Padmakar-Ivan indices of the strong product of graphs, Discuss. Math. Graph Theory, 32, pp. 749 - 769.
  • [27] Ramane, H. S. and Yalnaik, A.S., (2017), Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, J. Appl. Math. Comput., 55(1-2), pp. 609 - 627.
  • [28] Ramane, H.S., Yalnaik, A.S. and Sharafdini, R., (2018), Status connectivity indices and co-indices of graphs and its computation to some distance-balanced graph, AKCE Int. J. of Graphs and Comb., In Press.
  • [29] Sheeba Agnes, V., (2014), Degree distance of tensor product and strong product of graphs, Filomat, 28(10), pp. 2185 - 2198.
  • [30] Todeschini, R. and Consonni, V., (2009), Molecular Descriptors for Chemoinformatics, Wiley VCH, Weinheim.
  • [31] Wiener, H., (1947), Structural determination of paraffin boiling points., J. Am. Chem. Soc., 69, pp. 17 - 20.
  • [32] Xu, K., Liu, M., Das, K. C., Gutman, I. and Furtula, B., (2014), A survey on graphs extremal with respect to distance based topological indices, MATCH Commun. Math. Comput. Chem., 71, pp. 461 - 508.
  • [33] Yarahmadi, Z. and Ashrafi, A. R., (2012), The Szeged, vertex PI, first and second Zagreb indices of corona product of graphs, Filomat, 26, pp. 467 - 472.