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arXiv · 2510.19490

Closed-Form Analysis and Extremal Bounds of Albertson and Sigma Indices in Trees with Prescribed Degree Sequences

Jasem Hamoud·Alexey Belov Yakovlevich·Muaadh Almahalebi·Duaa Abdullah
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Abstract

This study explores the irregularity properties of trees with prescribed degree sequences by analyzing two prominent topological indices: the Albertson index and the sigma index. With a particular emphasis on caterpillar trees -frequently used to model molecular chains- we derive a closed-form expression for the Albertson index: \[ \mathrm{irr}(\mathscr{C}(n,m)) = m(m+1)n - 2m + 2, \quad \text{for } n \geq 3. \] Furthermore, we establish extremal bounds for both indices across tree families characterized by fixed degree sequences. The results yield a unified analytical framework for comparing linear and quadratic irregularity measures, and provide new structural insights relevant to applications in chemical graph theory and extremal graph analysis.

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Jasem Hamoud, Alexey Belov Yakovlevich, Muaadh Almahalebi, Duaa Abdullah. 2025-10-22. Closed-Form Analysis and Extremal Bounds of Albertson and Sigma Indices in Trees with Prescribed Degree Sequences. https://arxiv.org/abs/2510.19490

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