Extremizing antiregular graphs by modifying total σ-irregularity

Abstract

The total σ-irregularity is given by ${\sigma }_t\left(G\right)={\sum }_{\{u,v\}\subseteq V\left(G\right)}{\left(d_G\right(u)-d_G(v\left)\right)}^2$, where $d_G\left(z\right)$ 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 ${\sigma }_t^{f\left(n\right)}\left(G\right)={\sum }_{\{u,v\}\subseteq V\left(G\right)}\left|d_G\right(u)-d_G(v){|}^{f\left(n\right)}$, where $n=\left|V\right(G\left)\right|$ and $f\left(n\right)>0$. 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.