On a Conjecture on Edge Mostar Index of Bicyclic Graphs

Document Type : Research Paper

Authors

1 Department of Mathematics, Bishop Chulaparambil Memorial College, Kottayam-686001 & Department of Mathematics, Marthoma College, Pathanamthitta - 689103, India

2 Department of Mathematics, St. Aloysius College, Edathua, Alappuzha - 689573, India

Abstract

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

Keywords

Main Subjects


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