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On Mostar and Edge Mostar Indices of Graphs

Author

Listed:
  • Ali Ghalavand
  • Ali Reza Ashrafi
  • Mardjan Hakimi-Nezhaad
  • Ismail Naci Cangul

Abstract

Let G be a graph with edge set EG and e=uv∈EG. Define nue,G and mue,G to be the number of vertices of G closer to u than to v and the number of edges of G closer to u than to v, respectively. The numbers nve,G and mve,G can be defined in an analogous way. The Mostar and edge Mostar indices of G are new graph invariants defined as MoG=∑uv∈EGnuuv,G−nvuv,G and MoeG=∑uv∈EGmuuv,G−mvuv,G, respectively. In this paper, an upper bound for the Mostar and edge Mostar indices of a tree in terms of its diameter is given. Next, the trees with the smallest and the largest Mostar and edge Mostar indices are also given. Finally, a recent conjecture of Liu, Song, Xiao, and Tang (2020) on bicyclic graphs with a given order, for which extremal values of the edge Mostar index are attained, will be proved. In addition, some new open questions are presented.

Suggested Citation

  • Ali Ghalavand & Ali Reza Ashrafi & Mardjan Hakimi-Nezhaad & Ismail Naci Cangul, 2021. "On Mostar and Edge Mostar Indices of Graphs," Journal of Mathematics, Hindawi, vol. 2021, pages 1-14, April.
    • Handle: RePEc:hin:jjmath:6651220
    DOI: 10.1155/2021/6651220
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    Cited by:

    1. Hui Wang & Mengmeng Liu, 2023. "The Upper Bound of the Edge Mostar Index with Respect to Bicyclic Graphs," Mathematics, MDPI, vol. 11(11), pages 1-8, May.

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