Some New Results on Mostar Index of Graphs

Document Type : Research Paper


1 Department of mathematics, Shahid Rajaee Teacher Training University

2 Department of Mathematics, SRTT University

3 Department of Mathematics, Srtt University


A general bond additive index (GBA) can be defined as , where α(e) is edge contributions. The Mostar index is a new topological index whose edge contributions are α(e) = | nu - nv| in which nu is the number of vertices of lying closer to vertex u than to vertex v and nv can be defined similarly. In this paper, we propose some new results on the Mostar index based on the vertex-orbits under the action of automorphism group. In addition, we detrmined the structures of graphs with Mostar index equal 1. Finally, compute the Mostar index of a family of nanocone graphs.


  1. J. Bok, B. Furtula, N. Jedlickova and R. Skrekovski, On extremal graphs of weighted Szeged index, MATCH Commun. Math. Comput. Chem. 82 (2019) 93−109.
  2.  J. Devillers and A. T. Balaban (Eds.), Topological Indices and Related Descriptors in QSAR and QSPR, Amsterdam, Netherlands, Gordon and Breach, 2000.
  3. M. V. Diudea and I. Gutman, Wiener-type topological indices, Croat. Chem. Acta 71 (1) (1998) 21−51.
  4. D. Djoković, Distance preserving subgraphs of hypercubes, J. Combin. Theory Ser. B 14 (1973) 263−267.
  5. A. A. Dobrynin, On the Wiener index of certain families of fibonacenes, MATCH Commun. Math. Comput. Chem. 70 (2013) 565−574.
  6. A. A. Dobrynin, The Szeged and Wiener indices of line graphs, MATCH Commun. Math. Comput. Chem. 79 (2018) 743−756
  7. A. A. Dobrynin and I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math. Beograd. 56 (1994) 18−22.
  8. T. Došlić, I. Martinjak, R. Škrekovski, S. Tipurić Spužević and I. Zubac, Mostar index, J. Math. Chem. (2018), DOI:10.1007/s10910-018-0928-z.
  9. M. Ghorbani, X. Li, H. R. Maimani, Y. Mao, S. Rahmani and M. Rajabi-Parsa, Steiner (Revised) Szeged index of graphs, MATCH Commun. Math. Comput. Chem. 82 (2019) 733−742.
  10. M. Ghorbani; M. Songhori, Some topological indices of nanostar dendrimers, Iranian J. Math. Chem. 1 (2010) 57−65.
  11. M. Ghorbani and S. Rahmani, A note on Mostar index of a class of fullerenes, Int. J. of Chem. Model. 9 (2017) 245−256.
  12. I. Gutman, A formula for the Wiener number of trees and its extension to graphs containing cycles, Graph Theory Notes New York 27 (1994) 9−15.
  13. I. Gutman and A. A. Dobrynin, The Szeged index-a success story, Graph Theory Notes New York 34 (1998) 37−44.
  14. I. Gutman, P. V. Khadikar, P. V. Rajput and S. Karmarkar, The Szeged index of polyacenes, J. Serb. Chem. Soc. 60 (1995) 759−764.
  15. I. Gutman, S. Klavžar and B. Mohar (Eds), Fifty years of the Wiener index, MATCH Commun. Math. Comput. Chem. 35 (1997) 12−59.
  16. I. Gutman, Y. N. Yeh, S. L. Lee and Y. L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem. 32A (1993) 651−661.
  17. S. Klavžar and I. Gutman, A theorem on Wiener-type invariants for isometric subgraphs of hypercubes, Appl. Math. Lett. 19 (2006) 1129−1133.
  18. S. Klavžar, A. Rajapakse and I. Gutman, The Szeged and the Wiener index of graphs, Appl. Math. Lett. 9 (1996) 45−49.
  19. X. Li and M. Zhang, A note on the computation of revised (edge-)Szeged index in terms of canonical isometric embedding, MATCH Commun. Math. Comput. Chem. 81 (2019) 149−162.
  20. G. Rücker and C. Rücker, On topological indices, boiling points and cycloalkanes, J. Chem. Inf. Comput. Sci. 39 (1999) 788−802.
  21. B. E. Sagan, Y.-N. Yeh and P. Zhang, The Wiener polynomial of a graph, Int. J. Quantum. Chem. 60 (1996) 959−969.
  22. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley−VCH, Weinheim, 2000.
  23. H. J. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17−20.