Disproof of a conjecture on the edge Mostar index

Abstract

For a given connected graph $G$, the edge Mostar index $Mo_e(G)$ is defined as $Mo_e(G)=\sum_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|$, where $m_u(e|G)$ and $m_v(e|G)$ are respectively, the number of edges of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of edges of $G$ lying closer to vertex $v$ than to vertex $u$. We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that attain the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95--106].