Hyper-Zagreb indices of graphs and its applications

Girish V. Rajasekharaiah; Usha P. Murthy

doi:10.13069/jacodesmath.867532

Research Article

Hyper-Zagreb indices of graphs and its applications

Year 2021, Volume: 8 Issue: 1, 9 - 22, 15.01.2021

Girish V. Rajasekharaiah Usha P. Murthy

https://doi.org/10.13069/jacodesmath.867532

Cited By: 7

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] 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.

Year 2021, Volume: 8 Issue: 1, 9 - 22, 15.01.2021

Girish V. Rajasekharaiah Usha P. Murthy

https://doi.org/10.13069/jacodesmath.867532

Cited By: 7

Abstract

References

[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.

There are 34 citations in total.

Details

Primary Language	English
Subjects	Engineering
Journal Section	Articles
Authors	Girish V. Rajasekharaiah This is me 0000-0002-0036-6542 Usha P. Murthy This is me 0000-0001-9855-1887
Publication Date	January 15, 2021
Published in Issue	Year 2021 Volume: 8 Issue: 1

Cite

APA	Rajasekharaiah, G. V., & Murthy, U. P. (2021). Hyper-Zagreb indices of graphs and its applications. Journal of Algebra Combinatorics Discrete Structures and Applications, 8(1), 9-22. https://doi.org/10.13069/jacodesmath.867532
AMA	Rajasekharaiah GV, Murthy UP. Hyper-Zagreb indices of graphs and its applications. Journal of Algebra Combinatorics Discrete Structures and Applications. January 2021;8(1):9-22. doi:10.13069/jacodesmath.867532
Chicago	Rajasekharaiah, Girish V., and Usha P. Murthy. “Hyper-Zagreb Indices of Graphs and Its Applications”. Journal of Algebra Combinatorics Discrete Structures and Applications 8, no. 1 (January 2021): 9-22. https://doi.org/10.13069/jacodesmath.867532.
EndNote	Rajasekharaiah GV, Murthy UP (January 1, 2021) Hyper-Zagreb indices of graphs and its applications. Journal of Algebra Combinatorics Discrete Structures and Applications 8 1 9–22.
IEEE	G. V. Rajasekharaiah and U. P. Murthy, “Hyper-Zagreb indices of graphs and its applications”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 1, pp. 9–22, 2021, doi: 10.13069/jacodesmath.867532.
ISNAD	Rajasekharaiah, Girish V. - Murthy, Usha P. “Hyper-Zagreb Indices of Graphs and Its Applications”. Journal of Algebra Combinatorics Discrete Structures and Applications 8/1 (January 2021), 9-22. https://doi.org/10.13069/jacodesmath.867532.
JAMA	Rajasekharaiah GV, Murthy UP. Hyper-Zagreb indices of graphs and its applications. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8:9–22.
MLA	Rajasekharaiah, Girish V. and Usha P. Murthy. “Hyper-Zagreb Indices of Graphs and Its Applications”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 1, 2021, pp. 9-22, doi:10.13069/jacodesmath.867532.
Vancouver	Rajasekharaiah GV, Murthy UP. Hyper-Zagreb indices of graphs and its applications. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8(1):9-22.