Hyper-Zagreb indices of graphs and its applications

Abstract

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.

Keywords

• Degree of a vertex
• Hyper-Zagreb index
• Molecular graph

References

1. [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. [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. [3] F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, New York (1990)
4. [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. [5] K. C. Das, K. Xu, J. Nam, Zagreb indices of graphs, Front. Math. China 10 (2015) 567–582.
6. [6] K. C. Das, D. Lee, A. Graovac, Some properties of the Zagreb eccentricity indices, Ars Math. Contemp. 6 (2013) 117–125.
7. [7] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001) 211–249.
8. [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. [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. [10] I. Gutman, K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004) 83–92.
11. [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. [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. [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. [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. [15] F. Harary, Status and contrastatus, Sociometry 22 (1959) 23–43.
16. [16] F. Harary, Graph Theory, Narosa Publishing House, New Delhi (1999)
17. [17] H. Hua, K. C. Das, The relationship between the eccentric connectivity index and Zagreb indices, Discrete Appl. Math. 161 (2013) 2480–2491.
18. [18] A. Ilic, I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem. 65 (2011) 731–744.
19. [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. [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. [21] M. J. Morgan, S. Mukwembi, H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math. 311 (2011) 1229–1234.
22. [22] S. Nikolic, G. Kovacevic, A. Milicevic, N. Trinajstic, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003) 113–124.
23. [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. [24] S. Nikolic, N. Trinajstic, Z. Mihalic, The Wiener index: Development and applications, Croat. Chem. Acta 68 (1995) 105–129.
25. [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. [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. [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. [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. [29] R. Todeschini, Consonni, Handbook of Molecular Descriptors, Wiley, Weinheim (2000).
30. [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. [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. [32] H. Wiener, Structural determination of paraffin boiling point, J. Am. Chem. Soc. 69 (1947) 17–20.
33. [33] B. Zhou, I. Gutman, Further properties of Zagreb indices, MATCH Commun. Math. Comput. Chem. 54 (2005) 233–239.
34. [34] B. Zhou, Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem. 63 (2010) 181–198.