A Note on the Modified Albertson Index

Abstract

The modified Albertson index, denoted by $A{}^{\ast }$, of a graph $G$ is defined as $A{}^{\ast }(G)={\sum }_{uv\in E(G)}|(d_u{)}^2-(d_v{)}^2|$, 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 $A{}^{\ast }$ in terms of the maximum degree for the case of trees is derived. The $n$-vertex trees having maximal and minimal $A{}^{\ast }$ values are also characterized here. Moreover, it is shown that $A{}^{\ast }(G)$ is non-negative even integer for every graph $G$ and that there exist infinitely many connected graphs whose $A{}^{\ast }$ value is $2t$ for every integer $t\in \{0,3,4,5\}\cup \{8,9,10,\cdots \}$.