On a Conjecture on Edge Mostar Index of Bicyclic Graphs

Document Type : Research Paper

Authors

1 Department of Mathematics, Bishop Chulaparambil Memorial College, Kottayam-686001 & Department of Mathematics, Marthoma College, Pathanamthitta - 689103, India

2 Department of Mathematics, St. Aloysius College, Edathua, Alappuzha - 689573, India

Abstract

For an edge e = uv of a graph G, mu(e|G) denotes the number of edges closer
to the vertex u than to v (similarly mv(e|G)). The edge Mostar index Moe(G), of a graph
G is defined as the sum of absolute differences between mu(e|G) and mv(e|G) over all
edges e = uv of G. H. Liu et al. proposed a Conjecture on extremal bicyclic graphs with
respect to the edge Mostar index [1]. Even though the Conjecture was true in case of the
lower bound and proved in [2], it was wrong for the upper bound. In this paper, we
disprove the Conjecture proposed by H. Liu et al. [1], propose its correct version and
prove it. We also give an alternate proof for the lower bound of the edge Mostar index
for bicyclic graphs with a given number of vertices.

Keywords

Main Subjects

References

[1] H. Liu, L. Song, Q. Xiao and Z. Tang, On edge Mostar index of graphs, Iranian J. Math. Chem. 11 (2) (2020) 95–106, https://doi.org/10.22052/IJMC.2020.221320.1489.
[2] A. Ghalavand, A. R. Ashrafi and M. Hakimi-Nezhaad, On Mostar and edge Mostar indices of graphs, J. Math. (2021) ID 6651220, https://doi.org/10.1155/2021/6651220.
[3] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1) (1947) 17–20, https://doi.org/10.1021/ja01193a005.
[4] T. Došlic, I. Martinjak, R. Škrekovski, S. Tipuric Spuževic and I. Zubac, Mostar index, J. Math. Chem. 56 (2018) 2995–3013, https://doi.org/10.1007/s10910-018-0928-z.
[5] A. Ali and T. Došlic, Mostar index: results and perspectives, Appl. Math. Comput. 404 (2021) p. 126245, https://doi.org/10.1016/j.amc.2021.126245.
[6] F. Hayat and B. Zhou, On cacti with large Mostar index, Filomat 33 (15) (2019) 4865– 4873, https://doi.org/10.2298/FIL1915865H.
[7] S. Akhter, M. Imran and Z. Iqbal, Mostar indices of sio2 nanostructures and melem chain nanostructures, Int. J. Quantum Chem. 121 (5) (2021) e26520, https://doi.org/10.1002/qua.26520.
[8] N. Tratnik, Computing the Mostar index in networks with applications to molecular graphs, Iranian J. Math. Chem. 12 (1) (2021) 1–18, https://doi.org/10.22052/IJMC.2020.240316.1526.
[9] F. Gao, K. Xu and T. Došlic, On the difference of Mostar index and irregularity of graphs, Bull. Malays. Math. Sci. Soc. 44 (2021) 905–926, https://doi.org/10.1007/s40840-020-00991-y.
[10] 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 (1) (2021) 47–61.
[11] K. Deng and S. Li, Extremal Mostar indices of tree-like polyphenyls, Int. J. Quantum Chem. 121 (9) (2021) e26602, https://doi.org/10.1002/qua.26602.
[12] M. Arockiaraj, J. Clement and N. Tratnik, Mostar indices of carbon nanostructures and circumscribed donut benzenoid systems, Int. J. Quantum Chem. 119 (24) (2019) e26043,
https://doi.org/10.1002/qua.26043.
[13] M. Imran, S. Akhter and Z. Iqbal, Edge Mostar index of chemical structures and nanostructures using graph operations, Int. J. Quantum Chem. 120 (15) (2020) e26259, https://doi.org/10.1002/qua.26259.
[14] N. Ghanbari and S. Alikhani, Mostar index and edge Mostar index of polymers, Comput. Appl. Math. 40 (8) (2021) 1–21, https://doi.org/10.1007/s40314-021-01652-x.
[15] F. Yasmeen, S. Akhter, K. Ali and S. T. R. Rizvi, Edge Mostar indices of cacti graph with fixed cycles, Front. Chem. 9 (2021) p. 693885, https://doi.org/10.3389/fchem.2021.693885.
[16] A. Tepeh, Extremal bicyclic graphs with respect to Mostar index, Appl. Math. Comput. 355 (2019) 319–324, https://doi.org/10.1016/j.amc.2019.03.014.
Iranian Journal of Mathematical Chemistry
Volume 14, Issue 2
August 2023
Pages 97-108

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