A Note on the Modified Albertson Index

TL;DR

This paper investigates the properties of the modified Albertson index for graphs, establishing bounds for trees, characterizing extremal trees, and exploring the index's possible values across various graphs.

Contribution

It derives a sharp lower bound for the modified Albertson index in trees, characterizes extremal trees, and analyzes the index's value distribution in connected graphs.

Findings

A sharp lower bound of the index for trees is established.

Trees with maximal and minimal index values are characterized.

The index is always a non-negative even integer for any graph.

Abstract

The modified Albertson index, denoted by $A\!^*\!$, of a graph $G$ is defined as $A\!^*\!(G)=\sum_{uv\in E(G)} |(d_{u})^{2}- (d_{v})^{2}|$, where $d_u$, $d_v$ denote the degrees of the vertices $u$, $v$, respectively, of $G$ and $E(G)$ is the edge set of $G$. In this note, a sharp lower bound of $A\!^*$ in terms of the maximum degree for the case of trees is derived. The $n$-vertex trees having maximal and minimal $A\!^*$ values are also characterized here. Moreover, it is shown that $A\!^*\!(G)$ is non-negative even integer for every graph $G$ and that there exist infinitely many connected graphs whose $A\!^*$ value is $2t$ for every integer $t\in\{0,3,4,5\}\cup\{8,9,10,\cdots\}$.