Disproof of a Conjecture on the Edge Mostar Index

For a connected graph \(G\), the edge Mostar index \(Mo_e(G)\) is defined as \(Mo_e(G)=\sum\limits_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|\), 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 achieve the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95--106].