Extremizing antiregular graphs by modifying total $\sigma$-irregularity

The total $\sigma$-irregularity is given by ${\sigma }_t(G)={\sum }_{\{u,v\}\subseteq V(G)}{(d_G(u)-d_G(v))}^2,$ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing ${\sigma }_t$-irregularity are split graphs with only a few distinct degrees. Since one might typically expect that graphs with as many distinct degrees as possible achieve maximum irregularity measures, we modify this invariant to $\text{IR}(G)={\sum }_{\{u,v\}\subseteq V(G)}\mathrm{\mid }{d}_G(u)-d_G(v){\mathrm{\mid }}^{f(n)},$ where $n=\mathrm{\mid }V(G)\mathrm{\mid }$ and $f(n)>0$. We study under what conditions the above modification obtains its maximum for antiregular graphs. We consider general graphs, trees, and chemical graphs, and accompany our results with a few problems and conjectures.