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

2025/10/22 by Hamoud, Jasem, Yakovlevich, Alexey Belov, Almahalebi, Muaadh +1
#05C05 #05C12 #05C20 #05C25 #05C35 #05C76 #68R10 #Combinatorics (math.CO) #FOS: Mathematics #G.2.2

paper · doi:10.48550/arxiv.2510.19490

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: irr(\mathscrC(n,m)) = m(m+1)n - 2m + 2, for n ≥ 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.

Citations

• Several topological indices of random caterpillars
• The Number of Subtrees of Trees with Given Degree Sequence
• Asymptotics of trees with a prescribed degree sequence and applications
• A critical point for random graphs with a given degree sequence

Related

• Bounds on the Albertson Index for Trees with Given Degree Sequences
• Extremal Degree Irregularity Bounds for Albertson and Sigma Indices in Trees and Bipartite Graphs
• Degree Sequence of Albertson and σ-Indices on Trees of Order n\geqslant 3
• Regularities of Typical Sigma Index on Caterpillar Trees of Pendent Vertices
• On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds