On a Conjecture on Edge Mostar Index of Bicyclic Graphs

For an edge e = uv of a graph G, mu(e|G) denotes the number of edges closerto the vertex u than to v (similarly mv(e|G)). The edge Mostar index Moe(G), of a graphG is defined as the sum of absolute differences between mu(e|G) and mv(e|G) over alledges e = uv of G. H. Liu et al. proposed a Conjecture on extremal bicyclic graphs withrespect to the edge Mostar index [1]. Even though the Conjecture was true in case of thelower bound and proved in [2], it was wrong for the upper bound. In this paper, wedisprove the Conjecture proposed by H. Liu et al. [1], propose its correct version andprove it. We also give an alternate proof for the lower bound of the edge Mostar indexfor bicyclic graphs with a given number of vertices.