Counterexamples to a Conjecture on the Mostar Index of a Graph and its Line Graph

Document Type : Research Paper

Authors

1 Department of Applied Science‎, ‎Rajiv Gandhi Institute of Technology‎, ‎Kottayam-686501‎, ‎India

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

10.22052/ijmc.2026.257415.2052

Abstract

‎The Mostar index is a distance-based topological index defined as a quantitative measure of distance balancedness and peripherality in graphs‎. ‎Recently‎, ‎in “An inequality for the Mostar index of line graphs of trees” [Commun‎. ‎Comb‎. ‎Optim‎. ‎(2005)‎, ‎doi‎: ‎10.22049/cco.2025.30201.2356]‎, ‎the following conjecture regarding the Mostar index of graphs and their line graphs was proposed:
‎ Let G be a simple connected graph on $n$ vertices‎. ‎Then‎ Mo(L_G)<= Mo(G).
‎where‎, ‎L_G denotes the line graph of the graph G‎. ‎In this paper‎, ‎counterexamples are presented to disprove the conjecture proposed in \cite{sardar2025inequality}‎. ‎We also prove that the conjecture is not true for an infinite family of graphs with fixed cyclomatic number $c$‎, ‎where $1 \le c\le 3$‎.

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References

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Iranian Journal of Mathematical Chemistry
Volume 17, Issue 2
In Progress
June 2026
Pages 129-136

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