The general Albertson irregularity index of graphs

Zhen Lin; Ting Zhou; Xiaojing Wang; Lianying Miao; Zhen Lin; Ting Zhou; Xiaojing Wang; Lianying Miao

doi:10.3934/math.2022002

AIMS Mathematics

2022, Volume 7, Issue 1: 25-38. doi: 10.3934/math.2022002

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Research article

The general Albertson irregularity index of graphs

1.
School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, Qinghai, China
2.
School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, Jiangsu, China

Received: 19 August 2021 Accepted: 22 September 2021 Published: 29 September 2021
MSC : 05C05, 05C07, 05C09, 05C35

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.
- general Albertson irregularity index,
- tree,
- generalized Bethe tree,
- Kragujevac tree
Citation: Zhen Lin, Ting Zhou, Xiaojing Wang, Lianying Miao. The general Albertson irregularity index of graphs[J]. AIMS Mathematics, 2022, 7(1): 25-38. doi: 10.3934/math.2022002

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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.

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