On Mostar and Edge Mostar Indices of Graphs

Ali Ghalavand, Ali Reza Ashrafi, Mardjan Hakimi-Nezhaad and Ismail Naci Cangul

Journal of Mathematics, 2021, vol. 2021, 1-14

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.

Date: 2021
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:6651220

DOI: 10.1155/2021/6651220

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