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

TL;DR

This paper establishes extremal bounds for Albertson and Sigma indices in trees and bipartite graphs, providing new formulas and inequalities to understand their maximum and minimum irregularity measures.

Contribution

It introduces new bounds and explicit formulas for the Albertson and Sigma indices, enhancing understanding of extremal irregularity in bipartite graphs and trees.

Findings

Derived lower bounds for the Albertson index in bipartite graphs.

Established upper bounds for Albertson and Sigma indices based on graph parameters.

Provided examples illustrating the extremal irregularity bounds.

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.