Extremizing antiregular graphs by modifying total $\ddot{I}ƒ$-irregularity

Abstract

The total $\ddot{I}ƒ$-irregularity is given by $\ddot{I}{ƒ}_t(G)={\sum }_{\{u,v\}\subseteq V(G)}{\left(d_G(u)-d_G(v)\right)}^2,$ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing $\ddot{I}{ƒ}_t$-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 $ \IR(G)= \sum_{\{u,v\} \subseteq V(G)} |d_G(u)-d_G(v)|^{f(n)}, $ where $n=|V(G)|$ 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.

10 pages, 1 figure