Topological Indices Among Strong Support Vertex
Jasem Hamoud, Duaa Abdullah
TL;DR
The paper addresses irregularity-based indices for trees with strong support vertices, focusing on the Albertson index $irr(G)$ and the Sigma index $\sigma(G)$. It develops explicit degree-sequence–driven expressions and extremal bounds, showing how local pendant structures around strong support vertices control irregularity. Key contributions include conditional leaf-move transformations that reduce $irr$ and a degree-dependent behavior for $\sigma$, with caterpillar trees often achieving maximal $irr$ under these constraints. These results deepen understanding of how degree distributions around pendant structures shape topological indices in trees and offer guidance for designing trees with specific irregularity properties.
Abstract
In this paper, we provide the irregularity properties of trees with strong support vertex by analyzing two prominent topological indices: the Albertson index and the Sigma index. We further establish extremal bounds for both indices across families of trees defined by given degree sequences. Let $T_1$ and $T_2$ be a star trees of order $n$, where $T_1 \cong T_2$, then, we provide Albertson index of $T_1 \cong T_2$. Let $\mathcal{T}_{n, Δ}$ be a class of trees with $n$ vertices, there are a tree $T^{\prime} \in \mathcal{T}_{n, Δ}$ such that $\irr(T^{\prime}) < \irr(T)$.