Extremal Degree Irregularity Bounds for Albertson and Sigma Indices in Trees and Bipartite Graphs

Abstract

In this paper, the study of extreme value bounds for topological indices is crucial for understanding their influence on trees and bipartite graphs. For integers $\alpha, p$ satisfying $1 \leq p \leq \alpha \leq \Delta - 3$, the minimum Albertson index $\operatorname{irr}_{\min}$ has the following lower bound: $$ \operatorname{irr}_{\min} \geq \Delta^2 (\Delta - 1) \frac{2^{\alpha}}{(\Delta - p)^2}. $$ This work examines the maximum and minimum values of the Albertson index in bipartite graphs, depending on the sizes of their bipartition sets. Utilizing a known corollary, explicit formulas and bounds are derived to characterize the irregularity measure under various vertex size conditions. Additionally, upper bounds for the Albertson and Sigma indices are established using graph parameters such as degrees and edge counts. The theoretical findings are illustrated with examples and inequalities involving degree sequences, advancing the understanding of extremal irregularity behavior in bipartite graphs.