On Mostar and edge Mostar indices of graphs (Q2035723)

Summary: Let \(G\) be a graph with edge set \(E(G)\) and \(e=uv\in E(G)\). Define \(n_u (e,G)\) and \(m_u (e,G)\) 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 \(n_v (e,G)\) and \(m_v (e,G)\) can be defined in an analogous way. The Mostar and edge Mostar indices of \(G\) are new graph invariants defined as \(Mo (G)=\sum_{uv \in E(G)} |n_u (uv,G) - n_v(uv,G)|\) and \(Mo_e (G)=\sum_{u v \in E(G)} |m_u (uv,G) - m_v (uv,G)|\), 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 \textit{H. Liu} et al. [Iran. J. Math. Chem. 11, No. 2, 95--106 (2020; Zbl 1464.92310)] 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.