Note on Sharp Bounds for $\sigma$-Irregularity in Caterpillar Trees with Prescribed Degree Sequences

Determining sharp bounds for degree-based irregularity indices is a central problem in mathematical chemistry and extremal graph theory, with applications in QSAR/QSPR studies for predicting molecular properties.This paper examines the Albertson index and the Sigma index in trees, focusing on caterpillar trees $\mathbb{C}$ with given degree sequences $\mathscr{D}=(d_1,d_2,\dots,d_n)$.We derive sharp lower and upper bounds for the Sigma index, leveraging its relation to the Albertson index. Lower bounds characterize stable, low-irregularity structures, while upper bounds quantify maximal irregularity in highly asymmetric graphs; the quadratic Sigma index proves more sensitive in extreme cases.A fundamental result establishes that the degree sequence fully determines irregularity values in caterpillar trees, yielding closed-form expressions, precise extremal bounds, and characterizations of extremal trees. These results deepen the theoretical understanding of graph irregularity and improve predictive modeling of chemical structures.