An inequality for the Mostar index of line graphs of trees

Articles in Press

Document Type : Original paper

Authors

1 College of Mathematical Sciences, Harbin Engineering University, Harbin, PR China

2 Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia

3 Department of Mathematics Women University of Azad Jammu and Kashmir Bagh, India

4 General Administration of Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi Arabia

Abstract

Consider a simple connected graph G with the vertex set V (G) and edge set E(G). The Mostar index M◦(G) of G is defined as M◦(G) = 􏰌e=xy∈E(G) |nx − ny|, where nx and ny represent the number of vertices that lie closer to x than to y and the number of vertices that lie closer to y than to x, respectively. In this paper, we prove that if G is a tree, then M◦(LG) < M◦(G), where LG is the line graph. In order to provide an example supporting this result, we develop three algorithms (and implement them using Python) to calculate the Mostar index of trees of order at most 8 and their line graphs.

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References

[1] S. Akhter, Z. Iqbal, A. Aslam, and W. Gao, Computation of Mostar index for some graph operations, Int. J. Quantum Chem. 121 (2021), no. 15, Article ID: e26674.  https://doi.org/10.1002/qua.26674
[2] L. Alex and I. G, Sharp bounds on additively weighted Mostar index of cacti, Commun. Comb. Optim. (2024), In press.
https://doi.org/10.22049/cco.2024.28757.1702
[3] A. Ali and T. Došlić, Mostar index: Results and perspectives, Appl. Math. Comput. 404 (2021), Article ID: 126245. https://doi.org/10.1016/j.amc.2021.126245
[4] Y. Alizadeh, K. Xu, and S. Klavžar, On the Mostar index of trees and product graphs, Filomat 35 (2021), no. 14, 4637–4643.  https://doi.org/10.2298/FIL2114637A
[5] F. Asif, A. Kashif, S. Zafar, A. Aljaedi, and U. Albalawi, Mostar index of neural networks and applications, IEEE Access 11 (2023), 21345–21353.  https://doi.org/10.1109/ACCESS.2023.3246734
[6] S.H. Bertz, The bond graph, J. Chem. Soc. Chem. Commun. 16 (1981), 818–820. https://doi.org/10.1039/C39810000818
[7] S.H. Bertz, Branching in graphs and molecules, Discrete Appl. Math. 19 (1988), no. 1-3, 65–83. https://doi.org/10.1016/0166-218X(88)90006-6
[8] S.H. Bertz, A mathematical model of molecular complexity, Chemical Applications of Topology and Graph Theory (R.B. King, ed.), Elsevier, Amsterdam, 1993.
[9] B. Bollobás, Modern Graph Theory, Springer Science & Business Media, 2013.
[10] F. Buckley, Mean distance in line graphs, Congr. Numer. 32 (1981), 153–162.
[11] N. Dehgardi and M. Azari, More on mostar index, Appl. Math. E-Notes 20 (2020), 316–322.
[12] K. Deng and S. Li, Extremal catacondensed benzenoids with respect to the Mostar index, J. Math. Chem. 58 (2020), no. 7, 1437–1465.  https://doi.org/10.1007/s10910-020-01135-0
[13] K. Deng and S. Li, Chemical trees with extremal Mostar index, MATCH Commun. Math. Comput. Chem 85 (2021), no. 1, 161–180.
[14] T. Došlić, I. Martinjak, R. Škrekovski, S.T. Spužević, and I. Zubac, Mostar index, J. Math. Chem. 56 (2018), no. 10, 2995–3013.  https://doi.org/10.1007/s10910-018-0928-z
[15] M. Ghorbani, S. Rahmani, and M. Eslampoor, Some new results on Mostar index of graphs, Iran. J. Math. Chem. 11 (2020), no. 1, 33–42. https://doi.org/10.22052/ijmc.2020.209321.1475
[16] I. Gutman, A formula for the wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes N.Y. 27 (1994), no. 9, 9–15.
[17] I. Gutman, Distance of line graphs, Graph Theory Notes of New York (R.B. King, ed.), New York Academy of Sciences, Michigan, 1996.
[18] I. Gutman and S.J. Cyvin, Introduction to The theory of Benzenoid Hydrocarbons, Springer Science & Business Media, 2012.
[19] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), no. 4, 535–538. https://doi.org/10.1016/0009-2614(72)85099-1
[20] F. Harary, Line Graphs, Graph Theory, Addison-Wesley, Massachusetts, 1972, pp. 71–83.
[21] F. Harary, Graph Theory, New Age International, New Delhi, 2001.
[22] F. Hayat and B. Zhou, On cacti with large Mostar index, Filomat 33 (2019), no. 15, 4865–4873. https://doi.org/10.2298/FIL1915865H
[23] F. Hayat and B. Zhou, On Mostar index of trees with parameters, Filomat 33 (2019), no. 19, 6453–6458. https://doi.org/10.2298/FIL1919453H
[24] S. Huang, S. Li, and M. Zhang, On the extremal Mostar indices of hexagonal chains, MATCH Commun. Math. Comput. Chem 84 (2020), no. 1, 249–271.
[25] M. Imran, M.A. Malik, M. Aqib, G.I.H. Aslam, and A. Ali, On Zagreb coindices and Mostar index of TiO2 nanotubes, Sci. Rep. 13 (2023), no. 1, Article number: 13672.  https://doi.org/10.1038/s41598-023-40089-6
[26] J.B. Liu, S. Wang, C. Wang, and S. Hayat, Further results on computation of topological indices of certain networks, IET Control Theory Appl. 11 (2017), no. 13, 2065–2071.  https://doi.org/10.1049/iet–cta.2016.1237
[27] J.B. Liu and A. Zafari, Computing minimal doubly resolving sets and the strong metric dimension of the layer sun graph and the line graph of the layer sun graph, Complexity 2020 (2020), no. 1, Article ID: 6267072. https://doi.org/10.1155/2020/6267072
[28] J.B. Liu, J. Zhao, H. He, and Z. Shao, Valency-based topological descriptors and structural property of the generalized Sierpiński networks, J. Stat. Phys. 177 (2019), no. 6, 1131–1147. https://doi.org/10.1007/s10955-019-02412-2
[29] Z. Raza, N. ul Huda, F. Yasmeen, K. Ali, S. Akhter, and Y. Lin, Weighted Mostar invariants of chemical compounds: An analysis of structural stability, Heliyon 10 (2024), no. 10, Article ID: e30751. https://doi.org/10.1016/j.heliyon.2024.e30751
[30] P.M. Shihab, T.K.M. T. K. M. Varkey, V. Lijo, L.J. Teena, A. Riyas, and T.J.K. Rajesh, Neighborhood number-based topological indices of graphene, Polycycl. Aromat. Compd. 44 (2024), no. 4, 2733–2751. https://doi.org/10.1080/10406638.2023.2221762
[31] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947), no. 1, 17–20. https://doi.org/10.1021/ja01193a005
[32] Q. Xiao, M. Zeng, Z. Tang, H. Deng, and H. Hua, Hexagonal chains with the first three minimal Mostar indices, MATCH Commun. Math. Comput. Chem 85 (2021), no. 1, 47–61.
Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 10 February 2025

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