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

TL;DR

This paper explores how modifying the total $\sigma$-irregularity measure with a power function affects the extremal properties of antiregular graphs, providing new insights into their degree irregularity.

Contribution

It introduces a modified irregularity measure and investigates its maximum values specifically for antiregular graphs across various graph classes.

Findings

Modified irregularity measure can be maximized by antiregular graphs under certain conditions.

Results include characterizations for general graphs, trees, and chemical graphs.

The paper proposes open problems and conjectures related to the measure.

Abstract

The total $\sigma$-irregularity is given by $ \sigma_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 $\sigma_{t}$-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 $ \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.