Irregularity of Graphs respecting Degree Bounds

Abstract

Albertson defined the irregularity of a graph $G$ as $irr(G)=\sum _{uv\in E(G)}|d_G(u)-d_G(v)|$. For a graph $G$ with $n$ vertices, $m$ edges, maximum degree $\Delta$, and $d=\lfloor \frac{\Delta m}{\Delta n-m}\rfloor$, we show $$irr(G)\le d(d+1)n+\frac{1}{\Delta }\left({\Delta }^2-(2d+1)\Delta -d^2-d\right)m.$$