On the trees with given matching number and the modified first Zagreb connection index

Document Type : Research Paper

Authors

1 Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, B-Block, Faisal Town, Lahore, Pakistan

2 Department of Mathematics, Faculty of Science, University of Gujrat, Gujrat, Pakistan

10.22052/ijmc.2021.242169.1554

Abstract

The modified first Zagreb connection index ZC1 for a graph G is defined as ZC1 (G) = \sum v∈V (G) dvτv , where dv is the degree of the vertex v and τv denotes the connection number of v (that is, the number of vertices at the distance 2 from the vertex v). Let Tn,α be the class of trees with order n and matching number α such that n > 2α−1. In this paper, we obtain the lower bounds on the modified first Zagreb connection index of trees belonging to the class Tn,α, for 2α − 1 < n < 3α + 2.

Keywords


1. A. Ali, K. C. Das and S. Akhter, On the extremal graphs for second Zagreb index with fixed number of vertices and cyclomatic number, Miskolc Math. Notes, in press.
2. A. Ali and N. Trinajstić, A novel/old modification of the first Zagreb index,
Mol. Inform. 37 (2018) 1800008.
3. J. A. Bondy and U. S. R. Murty,
Graph Theory with Applications, Elsevier, New York, 1976.
4. B. Borovićanin, K. C. Das, B. Furtula and I. Gutman, Bounds for Zagreb indices, 
MATCH Commun. Math. Comput. Chem. 78 (2017) 17–100.
5. B. Basavanagoud and E. Chitra, On the leap Zagreb indices of generalized -pointline transformation graphs when z = 1,
Int. J. Math. Comb. 2 (2018) 44–66.
6. G. Ducoffe, R. Marinescu-Ghemeci, C. Obreja, A. Popa and R. M. Tache, Extremal graphs with respect to the modified first Zagreb connection index, Proceedings of the 16th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, CNAM Paris, France June 18 (2018) 65–68.
7. Z. Du, A. Ali and N. Trinajstić, Alkanes with the first three maximal/minimal modified first Zagreb connection indices, Mol. Inform. 38 (2019) 1800116.
8. J. Devillersm and A. T. Balaban (Eds
.), Topological Indices and Related Descriptors in QSAR and QSPAR, Gordon and Breach, Amsterdam, 2000.
9. J. C. Dearden, The Use of Topological Indices in QSAR and QSPR Modeling, Advances in QSAR Modeling,
Springer, Cham. (2017) 57–88.
10. N. Fatima, A. A. Bhatti, A. Ali and W. Gao, Zagreb connection indices of two dendrimer nanostars,
Acta Chem. Iasi 27 (2019) 1–14.
11. I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons,
Chem. Phys. Lett. 17 (1972) 535–538.
12. I. Gutman, B. Rušić, N. Trinajstić and C. F. Wilcox, Graph theory and molecular orbitals. XII. Acyclic polyenes,
J. Chem. Phys. 62 (1975) 3399–3405.
13. Y. Hou and J. Li, Bounds on the largest eigenvalues of trees with a given size of matching,
Linear Algebra Appl. 342 (2002) 203–217.
14. F. Harary,
Graph Theory, Addison-Wesley, Reading, MA 1969.
15. F. Javaid, M. K. Jamil and I. Tomescu, Extremal -generalized quasi unicyclic graphs with respect to first and second Zagreb indices,
Discrete Appl. Math. 270 (2019) 153–158.
16. S. Khalid, J. Kok, A. Ali and M. Bashir, Zagreb connection indices of Ti nanotubes, 
Chemistry: Bulgarian J. Sci. Edu. 27 (2018) 86–92.
17. S. Manzoor, N. Fatima, A. A, Bhatti and A. Ali, Zagreb connection indices of some nanostructures,
Acta Chem. Iasi 26 (2018) 169-180.
18. A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs,
Commun. Comb. Optim. 2 (2017) 99–117.
19. S. Noureen, A. Ali and A. A. Bhatti, On the extremal Zagreb indices of n-vertex chemical trees with fixed number of segments or branching vertices,
MATCH Commun. Math. Comput. Chem. 84 (2020) 513–534.
20. S. Noureen, A. A. Bhatti and A. Ali, Extremal trees for the modified first Zagreb connection index with fixed number of segments or vertices of degree 2,
J. Taibah Uni. Sci. 14 (2019) 31–37.
21. A. M. Naji and N. D. Soner, The first leap Zagreb index of some graph operations,
Int. J. Appl. Graph Theory 2 (2018) 7–18.
22. S. Noureen, A. A. Bhatti and A. Ali, Extremum modified first Zagreb connection index of -vertex trees with fixed number of pendant vertices,
Discrete Dyn. Nat. Soc. (2020) 3295342.
23. Z. Shao, I. Gutman, Z. Li, S. Wang and P. Wu, Leap Zagreb indices of trees and unicyclic graphs, Commun. Comb. Optim. 3 (2018) 179–194.
24. J. H. Tang, U. Ali, M. Javaid and K. Shabbir, Zagreb connection indices of subdivision and semi-total point operations on graphs,
J. Chem. (2019) 9846913.
25. R. Todeschini and V. Consonni,
Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000.
26. R. Todeschini and V. Consonni,
Molecular Descriptors for Chemoinformatics, Wiley–VCH, Weinheim, 2009.
27. A. Yurtas, M. Togan, V. Lokesha, I. N. Cangul and I. Gutman, Inverse problem for Zagreb indices,
J. Math. Chem. 57 (2019) 609–615.
28. F. Zhan, Y. Qiao and J. Cai, Relations between the first Zagreb index and spectral moment of graphs,
MATCH Commun. Math. Comput. Chem. 81 (2019) 383–392.
29. J. M. Zhu, N. Dehgardi and X. Li, The third leap Zagreb index for trees,
J. Chem. (2019) 9296401.