On the first three minimum Mostar indices of tree-like phenylenes

TL;DR

This paper investigates the minimal values of the Mostar index in tree-like phenylenes, identifying the first three smallest indices and characterizing the structures that attain these minima.

Contribution

It determines the first three minimum Mostar indices of tree-like phenylenes and characterizes all structures achieving these minimal values.

Findings

Identified the first three minimal Mostar indices for tree-like phenylenes.

Characterized all tree-like phenylenes that attain these minimal indices.

Provided numerical examples and discussion on the results.

Abstract

Let $G =(V_{G}, E_{G})$ be a simple connected graph with its vertex set $V_{G}$ and edge set $E_{G}$. The Mostar index $Mo(G)$ was defined as $Mo(G)=\sum\limits_{e=uv\in E(G)}|n_{u}-n_{v}|$, where $n_{u}$ (resp., $n_{v}$) is the number of vertices whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$). In this study, we determine the first three minimum Mostar indices of tree-like phenylenes and characterize all the tree-like phenylenes attaining these values. At last, we give some numerical examples and discussion.