Hyper-Zagreb indices of graphs and its applications
Hyper-Zagreb indices of graphs and its applications
The first and second Hyper-Zagreb index of a connected graph $G$ is defined by $HM_{1}(G)=\sum_{uv \in E(G)}[d(u)+d(v)]^{2}$ and $HM_{2}(G)=\sum_{uv \in E(G)}[d(u).d(v)]^{2}$. In this paper, the first and second Hyper-Zagreb indices of certain graphs are computed. Also the bounds for the first and second Hyper-Zagreb indices are determined. Further linear regression analysis of the degree based indices with the boiling points of benzenoid hydrocarbons is carried out. The linear model, based on the Hyper-Zagreb index, is better than the models corresponding to the other distance based indices.
___
- [1] A. R. Ashrafi, M. Ghorbani, Eccentric connectivity index of fullerenes. In: Gutman, I., Furtula, B.
(eds.) Novel Molecular Structure Descriptors–Theory and Applications II, Uni. Kragujevac, Kragujevac
(2010) 183–192.
- [2] A. R. Ashrafi, M. Saheli, M. Ghorbani, The eccentric connectivity index of nanotubes and nanotori,
J. Comput. Appl. Math. 235 (2011) 4561–4566
- [3] F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, New York (1990)
- [4] K. C. Das, I. Gutman, Estimating the Wiener index by means of number of vertices, number of edges
and diameter, MATCH Commun. Math. Comput. Chem. 64 (2010) 647–660.
- [5] K. C. Das, K. Xu, J. Nam, Zagreb indices of graphs, Front. Math. China 10 (2015) 567–582.
- [6] K. C. Das, D. Lee, A. Graovac, Some properties of the Zagreb eccentricity indices, Ars Math.
Contemp. 6 (2013) 117–125.
- [7] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta
Appl. Math. 66 (2001) 211–249.
- [8] T. Doslic, M. Saheli, Eccentric connectivity index of benzenoid graphs. In: Gutman, I., Furtula, B.
(eds.) Novel Molecular Structure Descriptors–Theory and Applications II, Uni. Kragujevac, Kragujevac
(2010) 169–183.
- [9] S. Gupta, M. Singh, A. K. Madan, Application of graph theory: relationship of eccentric connectivity
index and Wiener’s index with anti-inflammatory activity, J. Math. Anal. Appl. 266 (2002) 259–268.
- [10] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput.
Chem. 50 (2004) 83–92.
- [11] I. Gutman, B. Furtula, Z. Kovijani c Vukicevi, G. Popivoda, On Zagreb indices and coindices,
MATCH Commun. Math. Comput. Chem. 74 (2015) 5–16.
- [12] I. Gutman, B. Ruscic, N. Trinajstic, C. F. Wilcox, Graph theory and molecular orbitals, XII, acyclic
polyenes, J. Chem. Phys. 62 (1975) 3399–3405.
- [13] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals, Total Pi-electron energy of alternant
hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.
- [14] I. Gutman, Y. Yeh, S. Lee, Y. Luo, Some recent results in the theory of the Wiener number, Indian
J. Chem. 32A (1993) 651–661.
- [15] F. Harary, Status and contrastatus, Sociometry 22 (1959) 23–43.
- [16] F. Harary, Graph Theory, Narosa Publishing House, New Delhi (1999)
- [17] H. Hua, K. C. Das, The relationship between the eccentric connectivity index and Zagreb indices,
Discrete Appl. Math. 161 (2013) 2480–2491.
- [18] A. Ilic, I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput.
Chem. 65 (2011) 731–744.
- [19] M. H. Khalifeh, H. Yousefi-Azari, Ashrafi, The first and second Zagreb indices of some graph operations, Discrete Appl. Math. 157 (2009) 804–811.
- [20] V. Kumar, S. Sardana, A. K. Madan, Predicting anti-HIV activity of 2,3-diaryl-1, 3 thiazolidin-4-
ones: Computational approach using reformed eccentric connectivity index, J. Mol. Model 10 (2004)
399–407.
- [21] M. J. Morgan, S. Mukwembi, H. C. Swart, On the eccentric connectivity index of a graph, Discrete
Math. 311 (2011) 1229–1234.
- [22] S. Nikolic, G. Kovacevic, A. Milicevic, N. Trinajstic, The Zagreb indices 30 years after, Croat. Chem.
Acta 76 (2003) 113–124.
- [23] S. Nikolic, A. Milicevic, N. Trinajstic, A. Juric, On use of the variable Zagreb vM2 index in QSPR:
boiling points of benzenoid hydrocarbons, Molecules, 9 (2004) 1208–1221.
- [24] S. Nikolic, N. Trinajstic, Z. Mihalic, The Wiener index: Development and applications, Croat. Chem.
Acta 68 (1995) 105–129.
- [25] H. S. Ramane, V. V. Manjalapur, Note on the bounds on Wiener number of a graph, MATCH Commun. Math. Comput. Chem. 76 (2016) 19–22.
- [26] H. S. Ramane, D. S. Revankar, A. B. Ganagi, On the Wiener index of a graph, J. Indones. Math.
Soc. 18 (2012) 57–66.
- [27] H. S. Ramane, A. S. Yalnaik, Status connectivity indices of graphs and its applications to the boiling
point of benzenoid hydrocarbons, Journal of Applied Mathematics and Computing 55 (2017) 609–
627.
- [28] S. Sardana, A. K. Madan, Application of graph theory: Relationship of molecular connectivity
index, Wiener’s index and eccentric connectivity index with diuretic activity, MATCH Commun.
Math. Comput. Chem. 43 (2001) 85–98.
- [29] R. Todeschini, Consonni, Handbook of Molecular Descriptors, Wiley, Weinheim (2000).
- [30] D. Vukicevic, A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov. 57 (2010) 524–528.
- [31] H. B. Walikar, V. S. Shigehalli, H. S. Ramane, Bounds on the Wiener number of a graph, MATCH
Commun. Math. Comput. Chem. 50 (2004) 117–132.
- [32] H. Wiener, Structural determination of paraffin boiling point, J. Am. Chem. Soc. 69 (1947) 17–20.
- [33] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem.
54 (2005) 233–239.
- [34] B. Zhou, Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem. 63
(2010) 181–198.