The general Albertson irregularity index of graphs

TL;DR

This paper introduces a generalized irregularity index for graphs based on degree differences, providing bounds, extremal values, and formulas for specific tree classes, extending previous irregularity measures.

Contribution

It defines the general Albertson irregularity index, generalizes existing indices, and derives bounds and formulas for specific graph classes.

Findings

Established bounds for the index.

Determined extremal values for trees.

Provided formulas for Bethe and Kragujevac trees.

Abstract

We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_{p}(G) =(\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in $G$. The new index is not only generalization of the well-known Albertson irregularity index and $\sigma$-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Finally, we give the calculation formula of the general Albertson index of generalized Bethe trees and Kragujevac trees.