Irregularity of Graphs respecting Degree Bounds

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