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

TL;DR

This paper derives a closed-form expression for the Albertson index in caterpillar trees and establishes extremal bounds for Albertson and sigma indices in trees with fixed degree sequences, offering new insights into graph irregularity measures.

Contribution

It provides the first closed-form formula for the Albertson index in caterpillar trees and establishes extremal bounds for both indices in trees with prescribed degree sequences.

Findings

Closed-form expression for Albertson index in caterpillar trees.

Extremal bounds for Albertson and sigma indices in trees with fixed degree sequences.

Unified framework for analyzing irregularity measures in trees.

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.